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<title>Periodic function</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Periodic function</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Periodic_mapping" class="mw-redirect" title="Periodic mapping">periodic mapping</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Period length" redirects here; not to be confused with <a href="Repeating_decimal" title="Repeating decimal">repeating decimal</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Aperiodic" and "Non-periodic" redirect here. For other uses, see <a href="Aperiodic_(disambiguation)" class="mw-disambig" title="Aperiodic (disambiguation)">Aperiodic (disambiguation)</a>.</div>

<p>A <b>periodic function</b> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> that repeats its values at regular intervals. For example, the trigonometric functions, which are used to describe <a href="Wave" title="Wave">waves</a> and other repeating phenomena, are periodic. Many aspects of the natural world have periodic behavior, such as the <a href="Phases_of_the_Moon" class="mw-redirect" title="Phases of the Moon">phases of the Moon</a>, the swinging of a <a href="Pendulum" title="Pendulum">pendulum</a>, and the <a href="Human_heart" class="mw-redirect" title="Human heart">beating</a> of a heart.

</p><p>The length of the interval over which a periodic function repeats is called its <b><a href="Period_(physics)" class="mw-redirect" title="Period (physics)">period</a></b>. Any function that is not periodic is called <b>aperiodic</b>.

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<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<p>A function is defined as <b>periodic</b> if its values repeat at regular intervals. For example, the positions of the hands on a <a href="Clock" title="Clock">clock</a> display periodic behavior as they cycle through the same positions every 12 hours. This repeating interval is known as the <b>period</b>.

</p><p>More formally, a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is periodic if there exists a constant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> such that

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<annotation encoding="application/x-tex">{\displaystyle f(x+P)=f(x)}</annotation>

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</math></span><img src="./ef4363f96e151f0dea0dada371e1c68461fe1b8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.519ex; height:2.843ex;" alt="{\displaystyle f(x+P)=f(x)}" loading="lazy"></span></dd></dl>

<p>for all values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> in the <a href="Domain_of_a_function" title="Domain of a function">domain</a>. A <b>nonzero</b> constant <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> for which this condition holds is called a <b>period</b> of the function.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>If a period <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> exists, any integer multiple <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nP}">

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<annotation encoding="application/x-tex">{\displaystyle nP}</annotation>

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</math></span><img src="./a9a4ea84878846281ad19ebadcf5c5778249873d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.14ex; height:2.176ex;" alt="{\displaystyle nP}" loading="lazy"></span> (for a positive integer <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>) is also a period. If there is a <i>least positive</i> period, it is called the <b>fundamental period</b> (also <b>primitive period</b> or <b>basic period</b>).<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Often, "the" period of a function is used to refer to its fundamental period.

</p><p>Geometrically, a periodic function's graph exhibits <a href="Translational_symmetry" title="Translational symmetry">translational symmetry</a>. Its graph is <a href="Invariant_(mathematics)" title="Invariant (mathematics)">invariant</a> under <a href="Translation_(geometry)" title="Translation (geometry)">translation</a> in the <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

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<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

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</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>-direction by a distance of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

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</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>. This implies that the entire graph can be formed from copies of one particular portion, repeated at regular intervals.

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<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<p>Periodic behavior can be illustrated through both common, everyday examples and more formal mathematical functions.

</p>

<div class="mw-heading mw-heading3"><h3 id="Real-valued_functions">Real-valued functions</h3></div>

<p>Functions that map real numbers to real numbers can display periodicity, which is often visualized on a graph.

</p>

<div class="mw-heading mw-heading4"><h4 id="Sawtooth_wave">Sawtooth wave</h4></div>

<p>An example is the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> that represents the "<a href="Fractional_part" title="Fractional part">fractional part</a>" of its argument. Its period is 1. For instance,

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<annotation encoding="application/x-tex">{\displaystyle f(0.5)=f(1.5)=f(2.5)=\cdots =0.5}</annotation>

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</math></span><img src="./f3a873a271b52b12584035c3b35e5fc75d4a2ce7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.268ex; height:2.843ex;" alt="{\displaystyle f(0.5)=f(1.5)=f(2.5)=\cdots =0.5}" loading="lazy"></span></dd></dl>

<p>The graph of the function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is a <a href="Sawtooth_wave" title="Sawtooth wave">sawtooth wave</a>.

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<div class="mw-heading mw-heading4"><h4 id="Trigonometric_functions">Trigonometric functions</h4></div>

<p>The trigonometric functions are common examples of periodic functions. The <a href="Sine_function" class="mw-redirect" title="Sine function">sine function</a> and <a href="Cosine_function" class="mw-redirect" title="Cosine function">cosine function</a> are periodic with a fundamental period of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

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<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math></span><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"></span>, as illustrated in the figure to the right. For the sine function, this is expressed as:

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<annotation encoding="application/x-tex">{\displaystyle \sin(x+2\pi )=\sin x}</annotation>

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</math></span><img src="./efed452cc90f7cd14d656a2d97750c9e09a011a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19ex; height:2.843ex;" alt="{\displaystyle \sin(x+2\pi )=\sin x}" loading="lazy"></span></dd></dl>

<p>for all values of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>.

</p><p>The field of <a href="Fourier_series" title="Fourier series">Fourier series</a> investigates the concept that an arbitrary periodic function can be expressed as a sum of trigonometric functions with matching periods.

</p>

<div class="mw-heading mw-heading4"><h4 id="Exotic_functions">Exotic functions</h4></div>

<p>Some functions are periodic but possess properties that make them less intuitive. The <a href="Dirichlet_function" title="Dirichlet function">Dirichlet function</a>, for example, is periodic, with any nonzero rational number serving as a period. However, it does not possess a fundamental period.

</p>

<div class="mw-heading mw-heading3"><h3 id="Complex-valued_functions">Complex-valued functions</h3></div>

<p>Functions with a domain in the <a href="Complex_number" title="Complex number">complex numbers</a> can exhibit more complex periodic properties.

</p>

<div class="mw-heading mw-heading4"><h4 id="Complex_exponential">Complex exponential</h4></div>

<p>The complex exponential function is a periodic function with a purely imaginary period:

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<msup>

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<annotation encoding="application/x-tex">{\displaystyle e^{ikx}=\cos kx+i\,\sin kx}</annotation>

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</math></span><img src="./687d5865e24fc09078acb4efecdc03e677d929d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.018ex; height:2.843ex;" alt="{\displaystyle e^{ikx}=\cos kx+i\,\sin kx}" loading="lazy"></span></dd></dl>

<p>Given that the cosine and sine functions are both periodic with period <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

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</math></span><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"></span>, <a href="Euler's_formula" title="Euler's formula">Euler's formula</a> demonstrates that the complex exponential function has a period <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> such that

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</mfrac>

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<annotation encoding="application/x-tex">{\displaystyle L={\frac {2\pi }{k}}}</annotation>

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</math></span><img src="./d0fc93cc8e22c1e589004ac30771a55d63bd49da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:8.012ex; height:5.343ex;" alt="{\displaystyle L={\frac {2\pi }{k}}}" loading="lazy"></span>.</dd></dl>

<div class="mw-heading mw-heading4"><h4 id="Double-periodic_functions">Double-periodic functions</h4></div>

<p>A function on the complex plane can have two distinct, incommensurate periods without being a constant function. The <a href="Elliptic_function" title="Elliptic function">elliptic functions</a> are a primary example of such functions. ("Incommensurate" in this context refers to periods that are not real multiples of each other.)

</p>

<div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2></div>

<p>Periodic functions can take on values many times. More specifically, if a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is periodic with period <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>, then for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> in the domain of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> and all positive integers <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

</semantics>

</math></span><img src="./a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"></span>,<sup id="cite_ref-:1_3-0" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(x+nP)=f(x)}</annotation>

</semantics>

</math></span><img src="./f307bd27c269b7e8aec5133033cf143d38383683.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.914ex; height:2.843ex;" alt="{\displaystyle f(x+nP)=f(x)}" loading="lazy"></span></dd></dl>

<p>A significant property related to integration is that if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math></span><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span> is an <a href="Integral" title="Integral">integrable</a> periodic function with period <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>, then its definite integral over any interval of length <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> is the same.<sup id="cite_ref-:1_3-1" class="reference"><a href="#cite_note-:1-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> That is, for any real number <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span>:

</p>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{a}^{a+P}f(x)\,dx=\int _{0}^{P}f(x)\,dx}</annotation>

</semantics>

</math></span><img src="./cae9c43a68bd87dfd3fcafb577588cf300fc193f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:28.582ex; height:6.176ex;" alt="{\displaystyle \int _{a}^{a+P}f(x)\,dx=\int _{0}^{P}f(x)\,dx}" loading="lazy"></span></dd></dl>

<p>This property is crucial in areas such as <a href="Fourier_series" title="Fourier series">Fourier series</a>, where the coefficients are determined by integrals over one period.

</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math></span><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"></span> is a function with period <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>, then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(ax)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(ax)}</annotation>

</semantics>

</math></span><img src="./f6bef4b11bfd7449ad27f07af46e9a70c58d910c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.647ex; height:2.843ex;" alt="{\displaystyle f(ax)}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math></span><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"></span> is a non-zero real number such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ax}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle ax}</annotation>

</semantics>

</math></span><img src="./5d44662eeb8cbba7277da838b75c77d8cd3a4547.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.559ex; height:1.676ex;" alt="{\displaystyle ax}" loading="lazy"></span> is within the domain of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span>, is periodic with period <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {P}{|a|}}}">

<mfrac>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {P}{|a|}}}</annotation>

</semantics>

</math></span><img src="./4d58ba60594c880780b4f3be5614345c6254e239.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:3.36ex; height:6.009ex;" alt="{\displaystyle {\frac {P}{|a|}}}" loading="lazy"></span>. For example, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=\sin(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)=\sin(x)}</annotation>

</semantics>

</math></span><img src="./5fb1266b7f7718442e31e45eef3d81bef6a8b9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.511ex; height:2.843ex;" alt="{\displaystyle f(x)=\sin(x)}" loading="lazy"></span> has period <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math></span><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"></span> and, therefore, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sin(5x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sin(5x)}</annotation>

</semantics>

</math></span><img src="./cea6c23bbd59cce91b6cb2c5d581205ac72254df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.157ex; height:2.843ex;" alt="{\displaystyle \sin(5x)}" loading="lazy"></span> will have period <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {2\pi }{5}}}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {2\pi }{5}}}</annotation>

</semantics>

</math></span><img src="./23727b05ed3ed0d5484b6c5f0646c4480b040ab5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:3.331ex; height:5.176ex;" alt="{\displaystyle {\frac {2\pi }{5}}}" loading="lazy"></span>.

</p><p>A key property of many periodic functions is that they can be described by a <a href="Fourier_series" title="Fourier series">Fourier series</a>. This series represents a periodic function as a sum of simpler periodic functions, namely <a href="Sine_and_cosine" title="Sine and cosine">sines and cosines</a>. For example, a sound wave from a musical instrument can be broken down into the fundamental note and various <a href="Overtone" title="Overtone">overtones</a>. This decomposition is a powerful tool in fields like physics and signal processing. While most "well-behaved" periodic functions can be represented this way,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Fourier series can only be used for periodic functions or for functions defined on a finite length. If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> is a periodic function with period <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> that can be described by a Fourier series, the coefficients of the series can be described by an <a href="Integral" title="Integral">integral</a> over an interval of length <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>.

</p><p>Any function that is a combination of periodic functions with the same period is also periodic (though its fundamental period may be smaller). This includes:

</p>

<ul><li>addition, <a href="Subtraction" title="Subtraction">subtraction</a>, multiplication and division of periodic functions,<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> and</li>

<li>taking a power or a root of a periodic function (provided it is defined for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Generalizations">Generalizations</h2></div>

<p>The concept of periodicity can be generalized beyond functions on the real number line. For example, the idea of a repeating pattern can be applied to shapes in multiple dimensions, such as a periodic <a href="Tessellation" title="Tessellation">tessellation</a> of the plane. A <a href="Sequence_(mathematics)" class="mw-redirect" title="Sequence (mathematics)">sequence</a> can also be viewed as a function defined on the <a href="Natural_number" title="Natural number">natural numbers</a>, and the concept of a <a href="Periodic_sequence" title="Periodic sequence">periodic sequence</a> is defined accordingly.

</p>

<div class="mw-heading mw-heading3"><h3 id="Antiperiodic_functions">Antiperiodic functions</h3></div>

<p>One subset of periodic functions is that of <b>antiperiodic functions</b>. This is a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> such that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+P)=-f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x+P)=-f(x)}</annotation>

</semantics>

</math></span><img src="./893b17c172f171e8c84c22833d3e9a3df0a4556d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.328ex; height:2.843ex;" alt="{\displaystyle f(x+P)=-f(x)}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span>. For example, the sine and cosine functions are <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi }</annotation>

</semantics>

</math></span><img src="./9be4ba0bb8df3af72e90a0535fabcc17431e540a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.332ex; height:1.676ex;" alt="{\displaystyle \pi }" loading="lazy"></span>-antiperiodic and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\pi }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2\pi }</annotation>

</semantics>

</math></span><img src="./73efd1f6493490b058097060a572606d2c550a06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.494ex; height:2.176ex;" alt="{\displaystyle 2\pi }" loading="lazy"></span>-periodic. While a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span>-antiperiodic function is a <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 2P}</annotation>

</semantics>

</math></span><img src="./3b45c7ea79a10b769073cd75ff085612d3d13663.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.908ex; height:2.176ex;" alt="{\displaystyle 2P}" loading="lazy"></span>-periodic function, the <a href="Converse_(logic)" title="Converse (logic)">converse</a> is not necessarily true.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Bloch-periodic_functions">Bloch-periodic functions</h3></div>

<p>A further generalization appears in the context of <a href="Bloch's_theorem" title="Bloch's theorem">Bloch's theorems</a> and <a href="Floquet_theory" title="Floquet theory">Floquet theory</a>, which govern the solution of various periodic differential equations. In this context, the solution (in one dimension) is typically a function of the form

</p>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x+P)=e^{ikP}f(x)~,}</annotation>

</semantics>

</math></span><img src="./a685b1afd278e5f24b3a7f6ff2bb0b97c97e82cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.721ex; height:3.176ex;" alt="{\displaystyle f(x+P)=e^{ikP}f(x)~,}" loading="lazy"></span></dd></dl>

<p>where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math></span><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"></span> is a real or complex number (the <i>Bloch wavevector</i> or <i>Floquet exponent</i>). Functions of this form are sometimes called <b>Bloch-periodic</b> in this context. A periodic function is the special case <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=0}</annotation>

</semantics>

</math></span><img src="./6307c8a99dad7d0bcb712352ae0a748bd99a038b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}" loading="lazy"></span>, and an antiperiodic function is the special case <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=\pi /P}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=\pi /P}</annotation>

</semantics>

</math></span><img src="./4e20725e307518f83328d6b451dec2464408a7e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.55ex; height:2.843ex;" alt="{\displaystyle k=\pi /P}" loading="lazy"></span>. Whenever <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kP/\pi }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle kP/\pi }</annotation>

</semantics>

</math></span><img src="./10f37d649192ba12c439fec8dab160fed43b7a86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.451ex; height:2.843ex;" alt="{\displaystyle kP/\pi }" loading="lazy"></span> is rational, the function is also periodic.

</p>

<div class="mw-heading mw-heading3"><h3 id="Quotient_spaces_as_domain">Quotient spaces as domain</h3></div>

<p>In <a href="Signal_processing" title="Signal processing">signal processing</a> you encounter the problem, that <a href="Fourier_series" title="Fourier series">Fourier series</a> represent periodic functions and that Fourier series satisfy <a href="Convolution_theorem" title="Convolution theorem">convolution theorems</a> (i.e. <a href="Convolution" title="Convolution">convolution</a> of Fourier series corresponds to multiplication of represented periodic function and vice versa), but periodic functions cannot be convolved with the usual definition, since the involved integrals diverge. A possible way out is to define a periodic function on a bounded but periodic domain. To this end you can use the notion of a <a href="Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient space</a>:

</p>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathbb {R} /\mathbb {Z} }=\{x+\mathbb {Z} :x\in \mathbb {R} \}=\{\{y:y\in \mathbb {R} \land y-x\in \mathbb {Z} \}:x\in \mathbb {R} \}}</annotation>

</semantics>

</math></span><img src="./171611f51dffe578d90d4e5faa38539d43996584.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:59.92ex; height:2.843ex;" alt="{\displaystyle {\mathbb {R} /\mathbb {Z} }=\{x+\mathbb {Z} :x\in \mathbb {R} \}=\{\{y:y\in \mathbb {R} \land y-x\in \mathbb {Z} \}:x\in \mathbb {R} \}}" loading="lazy"></span>.</dd></dl>

<p>That is, each element in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathbb {R} /\mathbb {Z} }}">

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<annotation encoding="application/x-tex">{\displaystyle {\mathbb {R} /\mathbb {Z} }}</annotation>

</semantics>

</math></span><img src="./1fc347ef9fffe5ddc3bca496d27f9c3d7353de2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.391ex; height:2.843ex;" alt="{\displaystyle {\mathbb {R} /\mathbb {Z} }}" loading="lazy"></span> is an <a href="Equivalence_class" title="Equivalence class">equivalence class</a> of <a href="Real_number" title="Real number">real numbers</a> that share the same <a href="Fractional_part" title="Fractional part">fractional part</a>. Thus a function like <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:{\mathbb {R} /\mathbb {Z} }\to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle f:{\mathbb {R} /\mathbb {Z} }\to \mathbb {R} }</annotation>

</semantics>

</math></span><img src="./1d4745932417c3f41ba13e4c8724be7f2fb208b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.899ex; height:2.843ex;" alt="{\displaystyle f:{\mathbb {R} /\mathbb {Z} }\to \mathbb {R} }" loading="lazy"></span> is a representation of a 1-periodic function.

</p>

<div class="mw-heading mw-heading2"><h2 id="Calculating_period">Calculating period</h2></div>

<p>Consider a real waveform consisting of superimposed frequencies, expressed in a set as ratios to a <a href="Fundamental_frequency" title="Fundamental frequency">fundamental frequency</a>, f: F = <style data-mw-deduplicate="TemplateStyles:r1154941027">

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</style><span class="frac"><span class="num">1</span>⁄<span class="den">f</span></span> [f<sub>1</sub> f<sub>2</sub> f<sub>3</sub> ... f<sub>N</sub>] where all non-zero elements ≥1 and at least one of the elements of the set is 1. To find the period, T, first find the least common denominator of all the elements in the set. Period can be found as T = <span class="frac"><span class="num">LCD</span>⁄<span class="den">f</span></span>. Consider that for a simple sinusoid, T = <span class="frac"><span class="num">1</span>⁄<span class="den">f</span></span>. Therefore, the LCD can be seen as a periodicity multiplier.

</p>

<ul><li>For set representing all notes of Western <a href="Major_scale" title="Major scale">major scale</a>: [1 <span class="frac"><span class="num">9</span>⁄<span class="den">8</span></span> <span class="frac"><span class="num">5</span>⁄<span class="den">4</span></span> <span class="frac"><span class="num">4</span>⁄<span class="den">3</span></span> <span class="frac"><span class="num">3</span>⁄<span class="den">2</span></span> <span class="frac"><span class="num">5</span>⁄<span class="den">3</span></span> <span class="frac"><span class="num">15</span>⁄<span class="den">8</span></span>] the LCD is 24 therefore T = <span class="frac"><span class="num">24</span>⁄<span class="den">f</span></span>.</li>

<li>For set representing all notes of a major triad: [1 <span class="frac"><span class="num">5</span>⁄<span class="den">4</span></span> <span class="frac"><span class="num">3</span>⁄<span class="den">2</span></span>] the LCD is 4 therefore T = <span class="frac"><span class="num">4</span>⁄<span class="den">f</span></span>.</li>

<li>For set representing all notes of a minor triad: [1 <span class="frac"><span class="num">6</span>⁄<span class="den">5</span></span> <span class="frac"><span class="num">3</span>⁄<span class="den">2</span></span>] the LCD is 10 therefore T = <span class="frac"><span class="num">10</span>⁄<span class="den">f</span></span>.</li></ul>

<p>If no least common denominator exists, for instance if one of the above elements were irrational, then the wave would not be periodic.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

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<ul><li><a href="Almost_periodic_function" title="Almost periodic function">Almost periodic function</a></li>

<li><a href="Amplitude" title="Amplitude">Amplitude</a></li>

<li><a href="Continuous_wave" title="Continuous wave">Continuous wave</a></li>

<li><a href="Definite_pitch" class="mw-redirect" title="Definite pitch">Definite pitch</a></li>

<li><a href="Double_Fourier_sphere_method" title="Double Fourier sphere method">Double Fourier sphere method</a></li>

<li><a href="Doubly_periodic_function" title="Doubly periodic function">Doubly periodic function</a></li>

<li><a href="Fourier_transform" title="Fourier transform">Fourier transform</a> for computing periodicity in evenly spaced data</li>

<li><a href="Frequency" title="Frequency">Frequency</a></li>

<li><a href="Frequency_spectrum" class="mw-redirect" title="Frequency spectrum">Frequency spectrum</a></li>

<li><a href="Hill_differential_equation" title="Hill differential equation">Hill differential equation</a></li>

<li><a href="Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a> for computing periodicity in unevenly spaced data</li>

<li><a href="Periodic_sequence" title="Periodic sequence">Periodic sequence</a></li>

<li><a href="Periodic_summation" title="Periodic summation">Periodic summation</a></li>

<li><a href="Periodic_travelling_wave" title="Periodic travelling wave">Periodic travelling wave</a></li>

<li><a href="Quasiperiodic_function" title="Quasiperiodic function">Quasiperiodic function</a></li>

<li><a href="Seasonality" title="Seasonality">Seasonality</a></li>

<li><a href="Secular_variation" title="Secular variation">Secular variation</a></li>

<li><a href="Wavelength" title="Wavelength">Wavelength</a></li>

<li><a href="List_of_periodic_functions" title="List of periodic functions">List of periodic functions</a></li></ul></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

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</style><cite id="CITEREFTolstovTolstov2009" class="citation book cs1">Tolstov, Georgij Pavlovič; Tolstov, Georgij Pavlovič (2009). <i>Fourier series</i>. Dover books on mathematics (Nachdr. ed.). New York: Dover Publ. p. 1. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-63317-6</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">For some functions, like a <a href="Constant_function" title="Constant function">constant function</a> or the <a href="Dirichlet_function" title="Dirichlet function">Dirichlet function</a> (the <a href="Indicator_function" title="Indicator function">indicator function</a> of the <a href="Rational_number" title="Rational number">rational numbers</a>), a least positive period may not exist (the <a href="Infimum" class="mw-redirect" title="Infimum">infimum</a> of all positive periods <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math></span><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"></span> being zero).</span>

</li>

<li id="cite_note-:1-3"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_3-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_3-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFTolstov2009" class="citation book cs1">Tolstov, Georgij Pavlovič (2009). <i>Fourier series</i>. Dover books on mathematics (Nachdr. ed.). New York: Dover Publ. p. 2. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-63317-6</bdi>.</cite></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">For instance, for <a href="Lp_space" title="Lp space"><i>L</i><sup>2</sup> functions</a>, <a href="Carleson's_theorem" title="Carleson's theorem">Carleson's theorem</a> states that they have a <a href="Pointwise" title="Pointwise">pointwise</a> (<a href="Lebesgue_measure" title="Lebesgue measure">Lebesgue</a>) <a href="Almost_everywhere_convergence" class="mw-redirect" title="Almost everywhere convergence">almost everywhere convergent</a> <a href="Fourier_series" title="Fourier series">Fourier series</a>.</span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFWeisstein" class="citation web cs1">Weisstein, Eric W. <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/">"Antiperiodic Function"</a>. <i>mathworld.wolfram.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2024-06-06</span></span>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFSummerson2009" class="citation web cs1">Summerson, Samantha R. (5 October 2009). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20190825162000/https://www.ece.rice.edu/~srs1/files/Lec6.pdf">"Periodicity, Real Fourier Series, and Fourier Transforms"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="https://www.ece.rice.edu/~srs1/files/Lec6.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2019-08-25<span class="reference-accessdate">. Retrieved <span class="nowrap">2018-03-24</span></span>.</cite></span>

</li>

</ol></div></div>

<ul><li><cite id="CITEREFEkeland1990" class="citation book cs1"><a href="Ivar_Ekeland" title="Ivar Ekeland">Ekeland, Ivar</a> (1990). "One". <i>Convexity methods in Hamiltonian mechanics</i>. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)]. Vol. 19. Berlin: Springer-Verlag. pp. x+247. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-50613-6</bdi>. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1051888">1051888</a>.</cite></li></ul>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><cite class="citation cs1"><a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Periodic_function">"Periodic function"</a>. <i><a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a></i>. <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a>. 2001 [1994].</cite></li>

<li><span class="citation mathworld" id="Reference-Mathworld-Periodic_Function"><cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/PeriodicFunction.html">"Periodic Function"</a>. <i><a href="MathWorld" title="MathWorld">MathWorld</a></i>.</cite></span></li></ul>

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