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The `!rectangular function`! (also known as the `!rectangle function`!, `!rect function`!, `!Pi function`!, `!Heaviside Pi function`!,`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f] `!gate function`!, `!unit pulse`!, or the `!normalized `F33f`_`[boxcar function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Boxcar_function]`_`f`!) is defined as`:cite-ref-wolfram-2-0[`F5bf`_`[2`#cite-note-wolfram-2]`_`f]

rect ⁡ ⁡ ( t a ) = Π Π ( t a ) = { 0 , if | t | > a 2 1 2 , if | t | = a 2 1 , if | t | < a 2 . {\\displaystyle \\operatorname {rect} \\left({\\frac {t}{a}}\\right)=\\Pi \\left({\\frac {t}{a}}\\right)=\\left\\{{\\begin{array}{rl}0,&{\\text{if }}|t|>{\\frac {a}{2}}\\\\{\\frac {1}{2}},&{\\text{if }}|t|={\\frac {a}{2}}\\\\1,&{\\text{if }}|t|<{\\frac {a}{2}}.\\end{array}}\\right.}

Alternative definitions of the function define rect ⁡ ⁡ ( ± ± 1 2 ) {\\textstyle \\operatorname {rect} \\left(\\pm {\\frac {1}{2}}\\right)} to be 0,`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f] 1,`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f]`:cite-ref-5[`F5bf`_`[5`#cite-note-5]`_`f] or undefined.

Its periodic version is called a `*`F33f`_`[rectangular wave`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rectangular_wave]`_`f`*.

>>Contents

• `F0af`_`[History`#history]`_`f

• `F0af`_`[Relation to the boxcar function`#relation-to-the-boxcar-function]`_`f

• `F0af`_`[Fourier transform of the rectangular function`#fourier-transform-of-the-rectangular-function]`_`f

• `F0af`_`[Relation to the triangular function`#relation-to-the-triangular-function]`_`f

• `F0af`_`[Use in probability`#use-in-probability]`_`f

• `F0af`_`[Rational approximation`#rational-approximation]`_`f

• `F0af`_`[Demonstration of validity`#demonstration-of-validity]`_`f

• `F0af`_`[Dirac delta function`#dirac-delta-function]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

-─

>>History

The `*rect`* function has been introduced 1953 by `F33f`_`[Woodward`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Philip_Woodward]`_`f`:cite-ref-6[`F5bf`_`[6`#cite-note-6]`_`f] in "Probability and Information Theory, with Applications to Radar"`:cite-ref-7[`F5bf`_`[7`#cite-note-7]`_`f] as an ideal `F33f`_`[cutout operator`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Window_function]`_`f, together with the `F33f`_`[sinc function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sinc_function]`_`f`:cite-ref-8[`F5bf`_`[8`#cite-note-8]`_`f]`:cite-ref-9[`F5bf`_`[9`#cite-note-9]`_`f] as an ideal `F33f`_`[interpolation operator`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Whittaker–Shannon_interpolation_formula]`_`f, and their counter operations which are `F33f`_`[sampling`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sampling_(signal_processing)]`_`f (`F33f`_`[comb operator`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dirac_comb]`_`f) and `F33f`_`[replicating`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Periodic_summation]`_`f (`F33f`_`[rep operator`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dirac_comb]`_`f), respectively.

>>Relation to the boxcar function

The rectangular function is a special case of the more general `F33f`_`[boxcar function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Boxcar_function]`_`f:

rect ⁡ ⁡ ( t − − X Y ) = H ( t − − ( X − − Y / 2 ) ) − − H ( t − − ( X + Y / 2 ) ) = H ( t − − X + Y / 2 ) − − H ( t − − X − − Y / 2 ) {\\displaystyle \\operatorname {rect} \\left({\\frac {t-X}{Y}}\\right)=H(t-(X-Y/2))-H(t-(X+Y/2))=H(t-X+Y/2)-H(t-X-Y/2)}

where H ( x ) {\\displaystyle H(x)} is the `F33f`_`[Heaviside step function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Heaviside_step_function]`_`f; the function is centered at X {\\displaystyle X} and has duration Y {\\displaystyle Y} , from X − − Y / 2 {\\displaystyle X-Y/2} to X + Y / 2. {\\displaystyle X+Y/2.}

>>Fourier transform of the rectangular function

The `F33f`_`[unitary Fourier transforms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fourier_transform]`_`f of the rectangular function are`:cite-ref-wolfram-2-1[`F5bf`_`[2`#cite-note-wolfram-2]`_`f] ∫ ∫ − − ∞ ∞ ∞ ∞ rect ⁡ ⁡ ( t ) ⋅ ⋅ e − − i 2 π π f t d t = sin ⁡ ⁡ ( π π f ) π π f = sinc ⁡ ⁡ ( π π f ) = sinc π π ⁡ ⁡ ( f ) , {\\displaystyle \\int _{-\\infty }^{\\infty }\\operatorname {rect} (t)\\cdot e^{-i2\\pi ft}\\,dt={\\frac {\\sin(\\pi f)}{\\pi f}}=\\operatorname {sinc} (\\pi f)=\\operatorname {sinc} _{\\pi }(f),} using ordinary frequency f, where `F33f`_`[sinc π {\displaystyle \operatorname {sinc} _{\pi }}`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sinc_function]`_`f is the normalized form`:cite-ref-10[`F5bf`_`[10`#cite-note-10]`_`f] of the `F33f`_`[sinc function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sinc_function]`_`f and 1 2 π π ∫ ∫ − − ∞ ∞ ∞ ∞ rect ⁡ ⁡ ( t ) ⋅ ⋅ e − − i ω ω t d t = 1 2 π π ⋅ ⋅ sin ⁡ ⁡ ( ω ω / 2 ) ω ω / 2 = 1 2 π π ⋅ ⋅ sinc ⁡ ⁡ ( ω ω / 2 ) , {\\displaystyle {\\frac {1}{\\sqrt {2\\pi }}}\\int _{-\\infty }^{\\infty }\\operatorname {rect} (t)\\cdot e^{-i\\omega t}\\,dt={\\frac {1}{\\sqrt {2\\pi }}}\\cdot {\\frac {\\sin \\left(\\omega /2\\right)}{\\omega /2}}={\\frac {1}{\\sqrt {2\\pi }}}\\cdot \\operatorname {sinc} \\left(\\omega /2\\right),} using angular frequency ω ω {\\displaystyle \\omega } , where `F33f`_`[sinc {\displaystyle \operatorname {sinc} }`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sinc_function]`_`f is the unnormalized form of the `F33f`_`[sinc function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sinc_function]`_`f.

For rect ⁡ ⁡ ( x / a ) {\\displaystyle \\operatorname {rect} (x/a)} , its Fourier transform is ∫ ∫ − − ∞ ∞ ∞ ∞ rect ⁡ ⁡ ( t a ) ⋅ ⋅ e − − i 2 π π f t d t = a sin ⁡ ⁡ ( π π a f ) π π a f = a sinc π π ⁡ ⁡ ( a f ) . {\\displaystyle \\int _{-\\infty }^{\\infty }\\operatorname {rect} \\left({\\frac {t}{a}}\\right)\\cdot e^{-i2\\pi ft}\\,dt=a{\\frac {\\sin(\\pi af)}{\\pi af}}=a\\ \\operatorname {sinc} _{\\pi }{(af)}.}

>>Relation to the triangular function

We can define the `F33f`_`[triangular function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Triangular_function]`_`f as the `F33f`_`[convolution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Convolution]`_`f of two rectangular functions:

t r i ( t / T ) = r e c t ( 2 t / T ) ∗ ∗ r e c t ( 2 t / T ) . {\\displaystyle \\operatorname {tri(t/T)} =\\operatorname {rect(2t/T)} *\\operatorname {rect(2t/T)} .\\,}

>>Use in probability

Viewing the rectangular function as a `F33f`_`[probability density function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Probability_density_function]`_`f, it is a special case of the `F33f`_`[continuous uniform distribution`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Uniform_distribution_(continuous)]`_`f with a = − − 1 / 2 , b = 1 / 2. {\\displaystyle a=-1/2,b=1/2.} The `F33f`_`[characteristic function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Characteristic_function_(probability_theory)]`_`f is

φ φ ( k ) = sin ⁡ ⁡ ( k / 2 ) k / 2 , {\\displaystyle \\varphi (k)={\\frac {\\sin(k/2)}{k/2}},}

and its `F33f`_`[moment-generating function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Moment-generating_function]`_`f is

M ( k ) = sinh ⁡ ⁡ ( k / 2 ) k / 2 , {\\displaystyle M(k)={\\frac {\\sinh(k/2)}{k/2}},}

where sinh ⁡ ⁡ ( t ) {\\displaystyle \\sinh(t)} is the `F33f`_`[hyperbolic sine`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hyperbolic_sine]`_`f function.

>>Rational approximation

The pulse function may also be expressed as a limit of a `F33f`_`[rational function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rational_function]`_`f:

Π Π ( t ) = lim n → → ∞ ∞ , n ∈ ∈ ( Z ) 1 ( 2 t ) 2 n + 1 . {\\displaystyle \\Pi (t)=\\lim _{n\\rightarrow \\infty ,n\\in \\mathbb {(} Z)}{\\frac {1}{(2t)^{2n}+1}}.}

>>>Demonstration of validity

First, we consider the case where | t | < 1 2 . {\\textstyle |t|<{\\frac {1}{2}}.} Notice that the term ( 2 t ) 2 n {\\textstyle (2t)^{2n}} is always positive for integer n . {\\displaystyle n.} However, 2 t < 1 {\\displaystyle 2t<1} and hence ( 2 t ) 2 n {\\textstyle (2t)^{2n}} approaches zero for large n . {\\displaystyle n.}

It follows that: lim n → → ∞ ∞ , n ∈ ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = 1 0 + 1 = 1 , | t | < 1 2 . {\\displaystyle \\lim _{n\\rightarrow \\infty ,n\\in \\mathbb {(} Z)}{\\frac {1}{(2t)^{2n}+1}}={\\frac {1}{0+1}}=1,|t|<{\\tfrac {1}{2}}.}

Second, we consider the case where | t | > 1 2 . {\\textstyle |t|>{\\frac {1}{2}}.} Notice that the term ( 2 t ) 2 n {\\textstyle (2t)^{2n}} is always positive for integer n . {\\displaystyle n.} However, 2 t > 1 {\\displaystyle 2t>1} and hence ( 2 t ) 2 n {\\textstyle (2t)^{2n}} grows very large for large n . {\\displaystyle n.}

It follows that: lim n → → ∞ ∞ , n ∈ ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = 1 + ∞ ∞ + 1 = 0 , | t | > 1 2 . {\\displaystyle \\lim _{n\\rightarrow \\infty ,n\\in \\mathbb {(} Z)}{\\frac {1}{(2t)^{2n}+1}}={\\frac {1}{+\\infty +1}}=0,|t|>{\\tfrac {1}{2}}.}

Third, we consider the case where | t | = 1 2 . {\\textstyle |t|={\\frac {1}{2}}.} We may simply substitute in our equation:

lim n → → ∞ ∞ , n ∈ ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = lim n → → ∞ ∞ , n ∈ ∈ ( Z ) 1 1 2 n + 1 = 1 1 + 1 = 1 2 . {\\displaystyle \\lim _{n\\rightarrow \\infty ,n\\in \\mathbb {(} Z)}{\\frac {1}{(2t)^{2n}+1}}=\\lim _{n\\rightarrow \\infty ,n\\in \\mathbb {(} Z)}{\\frac {1}{1^{2n}+1}}={\\frac {1}{1+1}}={\\tfrac {1}{2}}.}

We see that it satisfies the definition of the pulse function. Therefore,

rect ⁡ ⁡ ( t ) = Π Π ( t ) = lim n → → ∞ ∞ , n ∈ ∈ ( Z ) 1 ( 2 t ) 2 n + 1 = { 0 if | t | > 1 2 1 2 if | t | = 1 2 1 if | t | < 1 2 . {\\displaystyle \\operatorname {rect} (t)=\\Pi (t)=\\lim _{n\\rightarrow \\infty ,n\\in \\mathbb {(} Z)}{\\frac {1}{(2t)^{2n}+1}}={\\begin{cases}0&{\\mbox{if }}|t|>{\\frac {1}{2}}\\\\{\\frac {1}{2}}&{\\mbox{if }}|t|={\\frac {1}{2}}\\\\1&{\\mbox{if }}|t|<{\\frac {1}{2}}.\\\\\\end{cases}}}

>>Dirac delta function

The rectangle function can be used to represent the `F33f`_`[Dirac delta function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dirac_delta_function]`_`f δ δ ( x ) {\\displaystyle \\delta (x)} .`:cite-ref-0-11-0[`F5bf`_`[11`#cite-note-0-11]`_`f] Specifically, δ δ ( x ) = lim a → → 0 1 a rect ⁡ ⁡ ( x a ) . {\\displaystyle \\delta (x)=\\lim _{a\\to 0}{\\frac {1}{a}}\\operatorname {rect} \\left({\\frac {x}{a}}\\right).} For a function g ( x ) {\\displaystyle g(x)} , its average over the width `* a {\\displaystyle a} `* around 0 in the function domain is calculated as,

g a v g ( 0 ) = 1 a ∫ ∫ − − ∞ ∞ ∞ ∞ d x g ( x ) rect ⁡ ⁡ ( x a ) . {\\displaystyle g_{avg}(0)={\\frac {1}{a}}\\int \\limits _{-\\infty }^{\\infty }dx\\ g(x)\\operatorname {rect} \\left({\\frac {x}{a}}\\right).} To obtain g ( 0 ) {\\displaystyle g(0)} , the following limit is applied,

g ( 0 ) = lim a → → 0 1 a ∫ ∫ − − ∞ ∞ ∞ ∞ d x g ( x ) rect ⁡ ⁡ ( x a ) {\\displaystyle g(0)=\\lim _{a\\to 0}{\\frac {1}{a}}\\int \\limits _{-\\infty }^{\\infty }dx\\ g(x)\\operatorname {rect} \\left({\\frac {x}{a}}\\right)} and this can be written in terms of the Dirac delta function as, g ( 0 ) = ∫ ∫ − − ∞ ∞ ∞ ∞ d x g ( x ) δ δ ( x ) . {\\displaystyle g(0)=\\int \\limits _{-\\infty }^{\\infty }dx\\ g(x)\\delta (x).} The Fourier transform of the Dirac delta function δ δ ( t ) {\\displaystyle \\delta (t)} is

δ δ ( f ) = ∫ ∫ − − ∞ ∞ ∞ ∞ δ δ ( t ) ⋅ ⋅ e − − i 2 π π f t d t = lim a → → 0 1 a ∫ ∫ − − ∞ ∞ ∞ ∞ rect ⁡ ⁡ ( t a ) ⋅ ⋅ e − − i 2 π π f t d t = lim a → → 0 sinc ⁡ ⁡ ( a f ) . {\\displaystyle \\delta (f)=\\int _{-\\infty }^{\\infty }\\delta (t)\\cdot e^{-i2\\pi ft}\\,dt=\\lim _{a\\to 0}{\\frac {1}{a}}\\int _{-\\infty }^{\\infty }\\operatorname {rect} \\left({\\frac {t}{a}}\\right)\\cdot e^{-i2\\pi ft}\\,dt=\\lim _{a\\to 0}\\operatorname {sinc} {(af)}.} where the `F33f`_`[sinc function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Sinc_function]`_`f here is the normalized sinc function. Because the first zero of the sinc function is at f = 1 / a {\\displaystyle f=1/a} and a {\\displaystyle a} goes to infinity, the Fourier transform of δ δ ( t ) {\\displaystyle \\delta (t)} is

δ δ ( f ) = 1 , {\\displaystyle \\delta (f)=1,} means that the frequency spectrum of the Dirac delta function is infinitely broad. As a pulse is shorten in time, it is larger in spectrum.

>>See also

• `F33f`_`[Fourier transform`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fourier_transform]`_`f

• `F33f`_`[Square wave`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Square_wave_(waveform)]`_`f

• `F33f`_`[Step function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Step_function]`_`f

• `F33f`_`[Top-hat filter`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Top-hat_filter]`_`f

• `F33f`_`[Boxcar function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Boxcar_function]`_`f

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `:citerefwolfram-research2008`aWolfram Research (2008). "HeavisidePi, Wolfram Language function". Retrieved October 11, 2022.

`:cite-note-wolfram-2`!2.`! `F0af`_`[↑`#cite-ref-wolfram-2-0]`_`f `:reference-mathworld-rectangle-function`a`:citerefweisstein`a`F33f`_`[Weisstein, Eric W.`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Eric_W._Weisstein]`_`f "Rectangle Function". `*`F33f`_`[MathWorld`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MathWorld]`_`f`*.

`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f `:citerefwang2012`aWang, Ruye (2012). `*Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis`*. Cambridge University Press. pp. 135–136. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9780521516884.

`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `:citereftang2007`aTang, K. T. (2007). `*Mathematical Methods for Engineers and Scientists: Fourier analysis, partial differential equations and variational models`*. Springer. p. 85. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9783540446958.

`:cite-note-5`!5.`! `F0af`_`[↑`#cite-ref-5]`_`f `:citerefkumar2011`aKumar, A. Anand (2011). `*Signals and Systems`*. PHI Learning Pvt. Ltd. pp. 258–260. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9788120343108.

`:cite-note-6`!6.`! `F0af`_`[↑`#cite-ref-6]`_`f `:citerefklauder1960`aKlauder, John R (1960). "The Theory and Design of Chirp Radars". `*Bell System Technical Journal`*. `!39`! (4): 745–808. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1002/j.1538-7305.1960.tb03942.x.

`:cite-note-7`!7.`! `F0af`_`[↑`#cite-ref-7]`_`f `:citerefwoodward1953`aWoodward, Philipp M (1953). `*Probability and Information Theory, with Applications to Radar`*. Pergamon Press. p. 29.

`:cite-note-8`!8.`! `F0af`_`[↑`#cite-ref-8]`_`f `:citerefhiggins1996`aHiggins, John Rowland (1996). `*Sampling Theory in Fourier and Signal Analysis: Foundations`*. Oxford University Press Inc. p. 4. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0198596995.

`:cite-note-9`!9.`! `F0af`_`[↑`#cite-ref-9]`_`f `:citerefzayed1996`aZayed, Ahmed I (1996). `*Handbook of Function and Generalized Function Transformations`*. CRC Press. p. 507. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 9780849380761.

`:cite-note-10`!10.`! `F0af`_`[↑`#cite-ref-10]`_`f Wolfram MathWorld, https://mathworld.wolfram.com/SincFunction.html

`:cite-note-0-11`!11.`! `F0af`_`[↑`#cite-ref-0-11-0]`_`f `:citerefkharebutolarajora2023`aKhare, Kedar; Butola, Mansi; Rajora, Sunaina (2023). "Chapter 2.4 Sampling by Averaging, Distributions and Delta Function". `*Fourier Optics and Computational Imaging`* (2nd ed.). Springer. pp. 15–16. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/978-3-031-18353-9. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-031-18353-9.

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