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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, `!elementary functions`! are those `F33f`_`[functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_(mathematics)]`_`f that are most commonly encountered by beginners. They are typically `F33f`_`[real functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Real_function]`_`f of a single real `F33f`_`[variable`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Variable_(mathematics)]`_`f that can be defined by applying the operations of `F33f`_`[addition`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Addition]`_`f, `F33f`_`[multiplication`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Multiplication]`_`f, `F33f`_`[division`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Division_(mathematics)]`_`f, `F33f`_`[nth root`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nth_root]`_`f, and `F33f`_`[function composition`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_composition]`_`f to `F33f`_`[polynomial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Polynomial_function]`_`f, `F33f`_`[exponential`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Exponential_function]`_`f, `F33f`_`[logarithm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithm]`_`f, and `F33f`_`[trigonometric`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Trigonometric_functions]`_`f functions. They include `F33f`_`[inverse trigonometric functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inverse_trigonometric_functions]`_`f, `F33f`_`[hyperbolic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hyperbolic_functions]`_`f and `F33f`_`[inverse hyperbolic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inverse_hyperbolic_functions]`_`f, which can be expressed in terms of logarithms and exponential function.

All elementary functions have `F33f`_`[derivatives`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Derivative]`_`f of any order, which are also elementary, and can be `F33f`_`[algorithmically`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algorithmically]`_`f computed by applying the `F33f`_`[differentiation rules`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Differentiation_rules]`_`f. The `F33f`_`[Taylor series`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Taylor_series]`_`f of an elementary function converges in a neighborhood of every point of its domain. More generally, they are `F33f`_`[global analytic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Global_analytic_function]`_`f, defined (possibly with multiple values, such as the elementary function z {\\displaystyle {\\sqrt {z}}} or log ⁡ ⁡ z {\\displaystyle \\log z} ) for every `F33f`_`[complex`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_number]`_`f argument, except at `F33f`_`[isolated points`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Isolated_point]`_`f. In contrast, `F33f`_`[antiderivatives`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Antiderivative]`_`f of elementary functions need not be elementary and is difficult to decide whether a specific elementary function has an elementary antiderivative.

In an attempt to solve this problem, `F33f`_`[Joseph Liouville`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Joseph_Liouville]`_`f introduced in 1833 a definition of elementary functions that extends the above one and is commonly accepted:`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f]`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f]`:cite-ref-3[`F5bf`_`[3`#cite-note-3]`_`f] An `*elementary function`* is a function that can be built, using addition, multiplication, division, and function composition, from `F33f`_`[constant functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Constant_function]`_`f, exponential functions, the `F33f`_`[complex logarithm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_logarithm]`_`f, and `F33f`_`[roots`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Polynomial_roots]`_`f of polynomials with elementary functions as coefficients. This includes the trigonometric functions, since, for example, ⁠ cos ⁡ ⁡ x = e i x + e − − i x 2 {\\displaystyle \\textstyle \\cos x={\\frac {e^{ix}+e^{-ix}}{2}}} ⁠, as well as every `F33f`_`[algebraic function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_function]`_`f.

Liouville's result is that, if an elementary function has an elementary antiderivative, then this antiderivative is a linear combination of logarithms, where the coefficients and the arguments of the logarithms are elementary functions involved, in some sense, in the definition of the function. More than 130 years later, `F33f`_`[Risch algorithm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Risch_algorithm]`_`f, named after `F33f`_`[Robert Henry Risch`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Robert_Henry_Risch]`_`f, is an algorithm to decide whether an elementary function has an elementary antiderivative, and, if it has, to compute this antiderivative. Despite dealing with elementary functions, the Risch algorithm is far from elementary; as of 2025, it seems that no complete implementation is available.

>>Contents

• `F0af`_`[Examples`#examples]`_`f
• `F0af`_`[Basic examples`#basic-examples]`_`f
• `F0af`_`[Composite examples`#composite-examples]`_`f
• `F0af`_`[Non-elementary functions`#non-elementary-functions]`_`f
• `F0af`_`[Closure`#closure]`_`f
• `F0af`_`[Differential algebra`#differential-algebra]`_`f
• `F0af`_`[See also`#see-also]`_`f
• `F0af`_`[Notes`#notes]`_`f
• `F0af`_`[References`#references]`_`f
• `F0af`_`[Further reading`#further-reading]`_`f
• `F0af`_`[External links`#external-links]`_`f

-─

>>Examples

>>>Basic examples

Elementary functions of a single variable x include:

• `F33f`_`[Constant functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Constant_function]`_`f: 2 , π π , e , {\\displaystyle 2,\\ \\pi ,\\ e,} the `F33f`_`[Euler–Mascheroni constant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Euler–Mascheroni_constant]`_`f, `F33f`_`[Apéry's constant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Apéry's_constant]`_`f, `F33f`_`[Khinchin's constant`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Khinchin's_constant]`_`f, etc. Any constant real (or complex) number.
• `F33f`_`[Powers of ⁠ x {\displaystyle x} ⁠`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Exponentiation]`_`f: x α α = e α α log ⁡ ⁡ x {\\displaystyle x^{\\alpha }=e^{\\alpha \\log x}} etc. (The exponent can be any real or complex constant.)
• `F33f`_`[Exponential functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Exponential_function]`_`f: e x , a x = e x log ⁡ ⁡ a {\\displaystyle \\textstyle e^{x},\\quad a^{x}=e^{x\\log a}}
• `F33f`_`[Logarithms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithm]`_`f: log ⁡ ⁡ x , log a ⁡ ⁡ x = log ⁡ ⁡ x log ⁡ ⁡ a {\\displaystyle \\textstyle \\log x,\\quad \\log _{a}x={\\frac {\\log x}{\\log a}}}
• `F33f`_`[Trigonometric functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Trigonometric_function]`_`f: sin ⁡ ⁡ x = e i x − − e − − i x 2 i , cos ⁡ ⁡ x = e i x + e − − i x 2 i , tan ⁡ ⁡ x = s i n x c o s x , {\\displaystyle \\textstyle \\sin x={\\frac {e^{ix}-e^{-ix}}{2i}},\\ \\cos x={\\frac {e^{ix}+e^{-ix}}{2i}},\\ \\tan x={\\frac {sinx}{cosx}},\\ } etc.
• `F33f`_`[Inverse trigonometric functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inverse_trigonometric_function]`_`f: arcsin ⁡ ⁡ x , arccos ⁡ ⁡ x , {\\displaystyle \\arcsin x,\\ \\arccos x,} etc.
• `F33f`_`[Hyperbolic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Hyperbolic_function]`_`f: sinh ⁡ ⁡ x , cosh ⁡ ⁡ x , {\\displaystyle \\sinh x,\\ \\cosh x,} etc.
• `F33f`_`[Inverse hyperbolic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inverse_hyperbolic_function]`_`f: arsinh ⁡ ⁡ x , arcosh ⁡ ⁡ x , {\\displaystyle \\operatorname {arsinh} x,\\ \\operatorname {arcosh} x,} etc.
• All functions obtained by adding, subtracting, multiplying or dividing a finite number of any of the previous functions`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f]
• All functions obtained as `F33f`_`[roots`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Root_of_a_polynomial]`_`f of a polynomial whose coefficients are elementary functions`:cite-ref-1-5-0[`F5bf`_`[5`#cite-note-1-5]`_`f]`:cite-ref-6[`F5bf`_`[6`#cite-note-6]`_`f]
• All functions obtained by `F33f`_`[composing`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_composition]`_`f a finite number of any of the previously listed functions

Certain elementary functions of a single complex variable z, such as z {\\displaystyle {\\sqrt {z}}} and log ⁡ ⁡ z {\\displaystyle \\log z} , may be `F33f`_`[multivalued`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Multivalued_function]`_`f. Additionally, certain classes of functions may be obtained by others using the final two rules. For example, the exponential function e z {\\displaystyle e^{z}} composed with addition, subtraction, and division provides the hyperbolic functions, while initial composition with i z {\\displaystyle iz} instead provides the trigonometric functions.

>>>Composite examples

Examples of elementary functions include:

• Addition, e.g. (x + 1)
• Multiplication, e.g. (2x)
• `F33f`_`[Polynomial`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Polynomial]`_`f functions
• e tan ⁡ ⁡ x 1 + x 2 sin ⁡ ⁡ ( 1 + ( log ⁡ ⁡ x ) 2 ) {\\displaystyle {\\frac {e^{\\tan x}}{1+x^{2}}}\\sin \\left({\\sqrt {1+(\\log x)^{2}}}\\right)}
• − − i log ⁡ ⁡ ( x + i 1 − − x 2 ) {\\displaystyle -i\\log \\left(x+i{\\sqrt {1-x^{2}}}\\right)}

The last function is equal to arccos ⁡ ⁡ x {\\displaystyle \\arccos x} , the `F33f`_`[inverse cosine`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Inverse_trigonometric_functions]`_`f, in the entire `F33f`_`[complex plane`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Complex_plane]`_`f.

All `F33f`_`[monomials`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Monomial]`_`f, `F33f`_`[polynomials`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Polynomial]`_`f, `F33f`_`[rational functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rational_function]`_`f and `F33f`_`[algebraic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_function]`_`f are elementary.

>>>Non-elementary functions

All elementary functions are `F33f`_`[analytic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Analytic_function]`_`f, unlike the `F33f`_`[absolute value function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Absolute_value]`_`f or discontinuous functions such as the `F33f`_`[step function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Step_function]`_`f.`:cite-ref-7[`F5bf`_`[7`#cite-note-7]`_`f]`:cite-ref-8[`F5bf`_`[8`#cite-note-8]`_`f] Some have proposed extending the set to include, for example, the `F33f`_`[Lambert W function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Lambert_W_function]`_`f`:cite-ref-9[`F5bf`_`[9`#cite-note-9]`_`f] or `F33f`_`[elliptic functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Elliptic_function]`_`f,`:cite-ref-10[`F5bf`_`[10`#cite-note-10]`_`f] all of which are still analytic.

Not every analytic function is elementary. Some examples that are `*not`* elementary, under standard definitions:

• `F33f`_`[tetration`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tetration]`_`f
• the `F33f`_`[gamma function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Gamma_function]`_`f
• non-elementary `F33f`_`[Liouvillian functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Liouvillian_function]`_`f, including

• the `F33f`_`[exponential integral`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Exponential_integral]`_`f (`*Ei`*), `F33f`_`[logarithmic integral`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Logarithmic_integral]`_`f (`*Li`* or `*li`*) and `F33f`_`[Fresnel integrals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Fresnel_integral]`_`f (`*S`* and `*C`*).
• the `F33f`_`[error function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Error_function]`_`f, e r f ( x ) = 2 π π ∫ ∫ 0 x e − − t 2 d t , {\\displaystyle \\mathrm {erf} (x)={\\frac {2}{\\sqrt {\\pi }}}\\int _{0}^{x}e^{-t^{2}}\\,dt,} a fact that may not be immediately obvious, but can be proven using the `F33f`_`[Risch algorithm`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Risch_algorithm]`_`f.

• other `F33f`_`[nonelementary integrals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nonelementary_integral]`_`f, including the `F33f`_`[Dirichlet integral`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Dirichlet_integral]`_`f and `F33f`_`[elliptic integral`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Elliptic_integral]`_`f.

>>Closure

It follows directly from the definition that the set of elementary functions is `F33f`_`[closed`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Closure_(mathematics)]`_`f under arithmetic operations, (algebraic) root extraction and composition. The elementary functions are closed under `F33f`_`[differentiation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Derivative]`_`f. They are not closed under `F33f`_`[limits and infinite sums`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Series_(mathematics)]`_`f. Importantly, the elementary functions are `*not`* closed under `F33f`_`[integration`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Antiderivative]`_`f, as shown by `F33f`_`[Liouville's theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Liouville's_theorem_(differential_algebra)]`_`f, see `F33f`_`[nonelementary integral`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Nonelementary_integral]`_`f. The `F33f`_`[Liouvillian functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Liouvillian_function]`_`f are defined as the elementary functions and, recursively, the integrals of the Liouvillian functions.

>>Differential algebra

The mathematical definition of an `!elementary function`!, or a function in elementary form, is considered in the context of `F33f`_`[differential algebra`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Differential_algebra]`_`f. A differential algebra is an algebra with the extra operation of derivation (algebraic version of differentiation). Using the derivation operation new equations can be written and their solutions used in `F33f`_`[extensions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Field_extension]`_`f of the algebra. By starting with the `F33f`_`[field`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Field_(mathematics)]`_`f of `F33f`_`[rational functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rational_function]`_`f, two special types of transcendental extensions (the logarithm and the exponential) can be added to the field building a tower containing elementary functions.

A `!differential field`! `*F`* is a field `*F`*0 (rational functions over the `F33f`_`[rationals`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rational_number]`_`f `!Q`! for example) together with a derivation map `*u`* → ∂`*u`*. (Here ∂`*u`* is a new function. Sometimes the notation `*u`*′ is used.) The derivation captures the properties of differentiation, so that for any two elements of the base field, the derivation is linear

∂ ∂ ( u + v ) = ∂ ∂ u + ∂ ∂ v {\\displaystyle \\partial (u+v)=\\partial u+\\partial v}

and satisfies the `F33f`_`[Leibniz product rule`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Product_rule]`_`f

∂ ∂ ( u ⋅ ⋅ v ) = ∂ ∂ u ⋅ ⋅ v + u ⋅ ⋅ ∂ ∂ v . {\\displaystyle \\partial (u\\cdot v)=\\partial u\\cdot v+u\\cdot \\partial v\\,.}

An element `*h`* is a constant if `*∂h = 0`*. If the base field is over the rationals, care must be taken when extending the field to add the needed transcendental constants.

A function `*u`* of a differential extension `*F`*[`*u`*] of a differential field `*F`* is an `!elementary function`! over `*F`* if the function `*u`*

• is `F33f`_`[algebraic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_function]`_`f over `*F`*, or
• is an `!exponential`!, that is, ∂`*u`* = `*u`* ∂`*a`* for `*a`* ∈ `*F`*, or
• is a `!logarithm`!, that is, ∂`*u`* = ∂`*a`* / a for `*a`* ∈ `*F`*.

(see also `F33f`_`[Liouville's theorem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Liouville's_theorem_(differential_algebra)]`_`f)

>>See also

• `F33f`_`[Algebraic function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Algebraic_function]`_`f
• `F33f`_`[Closed-form expression`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Closed-form_expression]`_`f
• `F33f`_`[Differential Galois theory`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Differential_Galois_theory]`_`f
• `F33f`_`[Elementary function arithmetic`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Elementary_function_arithmetic]`_`f
• `F33f`_`[Liouville's theorem (differential algebra)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Liouville's_theorem_(differential_algebra)]`_`f
• `F33f`_`[Tarski's high school algebra problem`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tarski's_high_school_algebra_problem]`_`f
• `F33f`_`[Transcendental function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Transcendental_function]`_`f
• `F33f`_`[Tupper's self-referential formula`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Tupper's_self-referential_formula]`_`f

>>Notes

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f `F33f`_`[Liouville 1833a`#citerefliouville1833a]`_`f.
`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `F33f`_`[Liouville 1833b`#citerefliouville1833b]`_`f.
`:cite-note-3`!3.`! `F0af`_`[↑`#cite-ref-3]`_`f `F33f`_`[Liouville 1833c`#citerefliouville1833c]`_`f.
`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `:citerefmorris-tenenbaum1985`aMorris Tenenbaum (1985). `*Ordinary Differential Equations`*. Dover. p. 17. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0-486-64940-7.
`:cite-note-1-5`!5.`! `F0af`_`[↑`#cite-ref-1-5-0]`_`f `:citerefspivak-michael-1994`aSpivak, Michael. (1994). `*Calculus`* (3rd ed.). Houston, Tex.: Publish or Perish. p. 363. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 0914098896. `F33f`_`[OCLC`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=OCLC_(identifier)]`_`f 31441929.
`:cite-note-6`!6.`! `F0af`_`[↑`#cite-ref-6]`_`f Ritt, chapter 1
`:cite-note-7`!7.`! `F0af`_`[↑`#cite-ref-7]`_`f `:citerefrisch1979`aRisch, Robert H. (1979). "Algebraic Properties of the Elementary Functions of Analysis". `*`F33f`_`[American Journal of Mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=American_Journal_of_Mathematics]`_`f`*. `!101`! (4): 743–759. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.2307/2373917. `F33f`_`[ISSN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISSN_(identifier)]`_`f 0002-9327. `F33f`_`[JSTOR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=JSTOR_(identifier)]`_`f 2373917.
`:cite-note-8`!8.`! `F0af`_`[↑`#cite-ref-8]`_`f Watson and Whittaker 1927, footnote to p 82
`:cite-note-9`!9.`! `F0af`_`[↑`#cite-ref-9]`_`f `:citerefstewart2005`aStewart, Seán (2005). "A new elementary function for our curricula?" (PDF). `*Australian Senior Mathematics Journal`*. `!19`! (2): 8–26.
`:cite-note-10`!10.`! `F0af`_`[↑`#cite-ref-10]`_`f Ince, footnote to p 330

>>References

• `:citerefliouville1833a`a`F33f`_`[Liouville, Joseph`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Joseph_Liouville]`_`f (1833a). "Premier mémoire sur la détermination des intégrales dont la valeur est algébrique". `*Journal de l'École Polytechnique`*. tome XIV: 124–148.
• `:citerefliouville1833b`a`F33f`_`[Liouville, Joseph`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Joseph_Liouville]`_`f (1833b). "Second mémoire sur la détermination des intégrales dont la valeur est algébrique". `*Journal de l'École Polytechnique`*. tome XIV: 149–193.
• `:citerefliouville1833c`a`F33f`_`[Liouville, Joseph`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Joseph_Liouville]`_`f (1833c). "Note sur la détermination des intégrales dont la valeur est algébrique". `*`F33f`_`[Journal für die reine und angewandte Mathematik`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Journal_für_die_reine_und_angewandte_Mathematik]`_`f`*. `!10`!: 347–359.
• `:citerefritt1950`a`F33f`_`[Ritt, Joseph`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Joseph_Ritt]`_`f (1950). `*Differential Algebra`*. `F33f`_`[AMS`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=American_Mathematical_Society]`_`f.
• `:citerefrosenlicht1972`a`F33f`_`[Rosenlicht, Maxwell`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Maxwell_Rosenlicht]`_`f (1972). "Integration in finite terms". `*`F33f`_`[American Mathematical Monthly`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=American_Mathematical_Monthly]`_`f`*. `!79`! (9): 963–972. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.2307/2318066. `F33f`_`[JSTOR`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=JSTOR_(identifier)]`_`f 2318066.

>>Further reading

• `:citerefdavenport2007`aDavenport, James H. (2007). "What Might "Understand a Function" Mean?". `*Towards Mechanized Mathematical Assistants`*. Lecture Notes in Computer Science. Vol. 4573. pp. 55–65. `F33f`_`[doi`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Doi_(identifier)]`_`f:10.1007/978-3-540-73086-6_5. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-3-540-73083-5. `F33f`_`[S2CID`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=S2CID_(identifier)]`_`f 8049737.

>>External links

• `*Elementary functions`* at Encyclopaedia of Mathematics
• `:reference-mathworld-elementary-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 "Elementary function". `*`F33f`_`[MathWorld`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=MathWorld]`_`f`*.

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