# Special Functions _z:complex_ Evaluate to the **error function** of a complex number. The error function is an odd function ( $$ \operatorname{erf} -z = - \operatorname{erf} z$$ ) that is used in statistics to calculate probabilities of normally distributed events. The formula for the error function of a complex number is: $$ \operatorname{erf} z = \frac{2}{\sqrt{\pi}} \int_0^z e^{-t^2} dt$$ where $$z$$ is a complex number. _z:complex_ Evaluate to the **complementary error function** of a complex number. It is defined as $$ \operatorname{erfc} z = 1 - \operatorname {erf} z $$. _x:real_ Evaluate to the **inverse error function** of a real number $$ -1 < x < 1 $$ It is defined as $$ \operatorname{erf} \left(\operatorname{erf} ^{-1}x\right) = x $$. _n_ ```json example ["Factorial", 5] // -> 120 ``` _n_ The double factorial of `n`: $$ n!! = n \cdot (n-2) \cdot (n-4) \times \cdots$$, that is the product of all the positive integers up to `n` that have the same parity (odd or even) as `n`. ```json example ["Factorial2", 5] // -> 15 ``` It can also be written in terms of the $$ \Gamma $$ function: $$ n!! = 2^{\frac{n}{2}+\frac{1}{4}(1-\cos(\pi n))}\pi^{\frac{1}{4}(\cos(\pi n)-1)}\Gamma\left(\frac{n}{2}+1\right) $$ This is not the same as the factorial of the factorial of `n` (i.e. $$((n!)!)$$). **Reference** - WikiPedia: [Double Factorial](https://en.wikipedia.org/wiki/Double_factorial) _z_ The [Gamma Function](https://en.wikipedia.org/wiki/Gamma_function) is an extension of the factorial function, with its argument shifted by 1, to real and complex numbers. $$ \operatorname{\Gamma}\left(z\right) = \int\limits_{0}^{\infty} t^{z-1} \mathrm{e}^{-t} \, \mathrm{d}t $$ - Wikidata: [Q190573](https://www.wikidata.org/wiki/Q190573) - NIST: http://dlmf.nist.gov/5.2.E1 ```json example ["Gamma", 5] // 24 ``` _s_, _z_ With two arguments, `Gamma` is the **upper incomplete Gamma function**. The lower limit of the integral is the second argument `z` instead of `0`: $$ \operatorname{\Gamma}\left(s, z\right) = \int\limits_{z}^{\infty} t^{s-1} \mathrm{e}^{-t} \, \mathrm{d}t $$ The order `s` and the lower limit `z` may be real or complex, including negative and fractional orders. The two-argument form is evaluated numerically with `.N()` and otherwise stays symbolic; $$ \Gamma(s, 0) $$ reduces to $$ \Gamma(s) $$. (This matches the `Gamma[s, z]` convention of Mathematica.) - NIST: http://dlmf.nist.gov/8.2.E2 ```json example ["N", ["Gamma", 2, 1]] // 0.7357588823428849 ``` _z_ This function is called `gammaln` in MatLab and SciPy and `LogGamma` in Mathematica. _s_ The [Riemann zeta function](https://en.wikipedia.org/wiki/Riemann_zeta_function), defined for complex numbers with real part greater than 1 as: $$ \zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} $$ and extended to other values by analytic continuation. ```json example ["Zeta", 2] // ➔ π²/6 ``` - Wikidata: [Q187235](https://www.wikidata.org/wiki/Q187235) - NIST: http://dlmf.nist.gov/25.2 _a_, _b_ The [Euler beta function](https://en.wikipedia.org/wiki/Beta_function), defined as: $$ \operatorname{B}(a, b) = \frac{\Gamma(a)\Gamma(b)}{\Gamma(a+b)} $$ It can also be expressed as an integral: $$ \operatorname{B}(a, b) = \int_0^1 t^{a-1}(1-t)^{b-1} \, dt $$ ```json example ["Beta", 2, 3] // ➔ 1/12 ``` - Wikidata: [Q192828](https://www.wikidata.org/wiki/Q192828) - NIST: http://dlmf.nist.gov/5.12 _a_, _z_ The **upper regularized incomplete gamma function**, defined as: $$ Q(a, z) = \frac{\Gamma(a, z)}{\Gamma(a)} = \frac{1}{\Gamma(a)}\int_z^\infty t^{a-1}e^{-t} \, dt $$ Exact arguments stay symbolic, and special values fold: `GammaRegularized(a, 0)` is $$1$$ and `GammaRegularized(1, z)` is $$e^{-z}$$. Use `N()` for a numeric value, at machine or arbitrary precision. The cumulative distribution function of a Poisson distribution evaluates to this function: $$\operatorname{CDF}(\operatorname{Poisson}(\lambda), k) = Q(k+1, \lambda)$$. ```json example ["N", ["GammaRegularized", 3, 5]] // ➔ 0.12465201948308115 ``` - NIST: http://dlmf.nist.gov/8.2 _x_, _a_, _b_ The **regularized incomplete beta function**, defined as: $$ I_x(a, b) = \frac{1}{\operatorname{B}(a, b)}\int_0^x t^{a-1}(1-t)^{b-1} \, dt $$ Exact arguments stay symbolic, and the endpoints fold: `BetaRegularized(0, a, b)` is $$0$$ and `BetaRegularized(1, a, b)` is $$1$$. Use `N()` for a numeric value, at machine or arbitrary precision. The cumulative distribution function of a binomial distribution evaluates to this function: $$\operatorname{CDF}(\operatorname{Binomial}(n, p), k) = I_{1-p}(n-k, k+1)$$. ```json example ["N", ["BetaRegularized", 0.5, 2, 3]] // ➔ 0.6875 ``` - NIST: http://dlmf.nist.gov/8.17 _x_ The [Lambert W function](https://en.wikipedia.org/wiki/Lambert_W_function), also called the product logarithm. It is the inverse function of $$ f(w) = w e^w $$. For a given value $x$, $W(x)$ is the value $w$ such that $w e^w = x$. The derivative of the Lambert W function is: $$ \frac{d}{dx} W(x) = \frac{W(x)}{x(1 + W(x))} $$ ```json example ["LambertW", 1] // ➔ Ω ≈ 0.5671 (the Omega constant) ``` - Wikidata: [Q429963](https://www.wikidata.org/wiki/Q429963) - NIST: http://dlmf.nist.gov/4.13 ## Fresnel Integrals The [Fresnel integrals](https://en.wikipedia.org/wiki/Fresnel_integral) arise in the description of near-field diffraction and in the geometry of the Cornu spiral (Euler spiral). _x_ The Fresnel S integral: $$ S(x) = \int_0^x \sin\!\left(\frac{\pi t^2}{2}\right) dt $$ It is an odd function ($S(-x) = -S(x)$) with asymptotic value $S(\infty) = \tfrac{1}{2}$. ```json example ["FresnelS", 1] // ➔ 0.4383 ``` - Wikipedia: [Fresnel integral](https://en.wikipedia.org/wiki/Fresnel_integral) - NIST: http://dlmf.nist.gov/7.2 _x_ The Fresnel C integral: $$ C(x) = \int_0^x \cos\!\left(\frac{\pi t^2}{2}\right) dt $$ It is an odd function ($C(-x) = -C(x)$) with asymptotic value $C(\infty) = \tfrac{1}{2}$. ```json example ["FresnelC", 1] // ➔ 0.7799 ``` - Wikipedia: [Fresnel integral](https://en.wikipedia.org/wiki/Fresnel_integral) - NIST: http://dlmf.nist.gov/7.2 ## Bessel Functions Bessel functions are solutions to Bessel's differential equation: $$ x^2 \frac{d^2 y}{dx^2} + x \frac{dy}{dx} + (x^2 - n^2)y = 0 $$ They arise in problems with cylindrical or spherical symmetry. _n_, _x_ The [Bessel function of the first kind](https://en.wikipedia.org/wiki/Bessel_function#Bessel_functions_of_the_first_kind) of order $n$. The derivative with respect to $x$ is: $$ \frac{d}{dx} J_n(x) = \frac{1}{2}(J_{n-1}(x) - J_{n+1}(x)) $$ ```json example ["BesselJ", 0, 1] // ➔ J₀(1) ≈ 0.7652 ``` - NIST: http://dlmf.nist.gov/10.2 _n_, _x_ The [Bessel function of the second kind](https://en.wikipedia.org/wiki/Bessel_function#Bessel_functions_of_the_second_kind) of order $n$, also called the Neumann function. The derivative with respect to $x$ is: $$ \frac{d}{dx} Y_n(x) = \frac{1}{2}(Y_{n-1}(x) - Y_{n+1}(x)) $$ ```json example ["BesselY", 0, 1] // ➔ Y₀(1) ≈ 0.0883 ``` - NIST: http://dlmf.nist.gov/10.2 _n_, _x_ The [modified Bessel function of the first kind](https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions) of order $n$. The derivative with respect to $x$ is: $$ \frac{d}{dx} I_n(x) = \frac{1}{2}(I_{n-1}(x) + I_{n+1}(x)) $$ ```json example ["BesselI", 0, 1] // ➔ I₀(1) ≈ 1.2661 ``` - NIST: http://dlmf.nist.gov/10.25 _n_, _x_ The [modified Bessel function of the second kind](https://en.wikipedia.org/wiki/Bessel_function#Modified_Bessel_functions) of order $n$, also called the MacDonald function. The derivative with respect to $x$ is: $$ \frac{d}{dx} K_n(x) = -\frac{1}{2}(K_{n-1}(x) + K_{n+1}(x)) $$ ```json example ["BesselK", 0, 1] // ➔ K₀(1) ≈ 0.4210 ``` - NIST: http://dlmf.nist.gov/10.25 ## Airy Functions Airy functions are solutions to the Airy differential equation: $$ \frac{d^2 y}{dx^2} - xy = 0 $$ They arise in physics, particularly in quantum mechanics and optics. _x_ The [Airy function of the first kind](https://en.wikipedia.org/wiki/Airy_function). It is the solution to the Airy equation that decays exponentially for positive $x$ and oscillates for negative $x$. ```json example ["AiryAi", 0] // ➔ 1/(3^(2/3) Γ(2/3)) ≈ 0.3550 ``` - NIST: http://dlmf.nist.gov/9.2 _x_ The [Airy function of the second kind](https://en.wikipedia.org/wiki/Airy_function). It is the solution to the Airy equation that grows exponentially for positive $x$ and oscillates for negative $x$. ```json example ["AiryBi", 0] // ➔ 1/(3^(1/6) Γ(2/3)) ≈ 0.6149 ``` - NIST: http://dlmf.nist.gov/9.2 _x_ The derivative of the [Airy function of the first kind](https://en.wikipedia.org/wiki/Airy_function). It satisfies $\operatorname{Ai}''(x) = x\operatorname{Ai}(x)$, so differentiating an Airy expression stays within the Airy family. ```json example ["AiryAiPrime", 0] // ➔ −1/(3^(1/3) Γ(1/3)) ≈ −0.2588 ``` - NIST: http://dlmf.nist.gov/9.2 _x_ The derivative of the [Airy function of the second kind](https://en.wikipedia.org/wiki/Airy_function). It satisfies $\operatorname{Bi}''(x) = x\operatorname{Bi}(x)$. ```json example ["AiryBiPrime", 0] // ➔ 3^(1/6)/Γ(1/3) ≈ 0.4483 ``` - NIST: http://dlmf.nist.gov/9.2 ## Elliptic Integrals The complete elliptic integrals arise in computing the arc length of an ellipse, the period of a pendulum, and throughout number theory. **Convention:** these functions use the **parameter** $m = k^2$, where $k$ is the modulus. This matches Mathematica, mpmath and [Fungrim](https://fungrim.org). To evaluate in terms of the modulus $k$, pass $k^2$. _m_ The [complete elliptic integral of the first kind](https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_first_kind): $$ K(m) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1 - m \sin^2\theta}} $$ It is computed via the arithmetic-geometric mean: $K(m) = \dfrac{\pi}{2\operatorname{agm}(1, \sqrt{1-m})}$. $K(1) = \infty$. For $m > 1$ the value is complex. ```json example ["EllipticK", 0.5] // ➔ 1.85407467730137 ``` - Wikidata: [Q1080993](https://www.wikidata.org/wiki/Q1080993) - NIST: http://dlmf.nist.gov/19.2 _m_ The [complete elliptic integral of the second kind](https://en.wikipedia.org/wiki/Elliptic_integral#Complete_elliptic_integral_of_the_second_kind): $$ E(m) = \int_0^{\pi/2} \sqrt{1 - m \sin^2\theta} \, d\theta $$ The perimeter of an ellipse with semi-major axis $a$ and eccentricity $e$ is $4aE(e^2)$. $E(1) = 1$. For $m > 1$ the value is complex. ```json example ["EllipticE", 0.5] // ➔ 1.35064388104768 ``` - Wikidata: [Q1375529](https://www.wikidata.org/wiki/Q1375529) - NIST: http://dlmf.nist.gov/19.2 _a_, _b_ _z_ The [arithmetic-geometric mean](https://en.wikipedia.org/wiki/Arithmetic%E2%80%93geometric_mean) of two numbers: the common limit of the sequences $a_{n+1} = \frac{a_n + b_n}{2}$ and $b_{n+1} = \sqrt{a_n b_n}$. With a single argument, $\operatorname{agm}(z)$ is shorthand for $\operatorname{agm}(1, z)$. ```json example ["AGM", 1, 2] // ➔ 1.45679103104691 ``` - Wikidata: [Q360074](https://www.wikidata.org/wiki/Q360074) - NIST: http://dlmf.nist.gov/19.8 ## Hypergeometric Functions _a_, _b_, _c_, _z_ The [Gauss hypergeometric function](https://en.wikipedia.org/wiki/Hypergeometric_function), defined for $|z| < 1$ by the series: $$ {}_2F_1(a, b; c; z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!} $$ where $(q)_n$ is the Pochhammer symbol (rising factorial), and extended elsewhere by analytic continuation. Many elementary and special functions are particular cases. For example $\ln(1+z) = z \cdot {}_2F_1(1, 1; 2; -z)$ and $K(m) = \frac{\pi}{2} \, {}_2F_1\big(\frac12, \frac12; 1; m\big)$. If $a$ or $b$ is a non-positive integer the series terminates and the function is a polynomial in $z$, evaluable for any $z$. Otherwise the function evaluates numerically for $z \le 1$ (real) and for complex $z$ within the unit disk and the Pfaff-transform region. ```json example ["Hypergeometric2F1", 1, 1, 2, 0.5] // ➔ 1.38629436111989 (= 2 ln 2) ``` - Wikidata: [Q672619](https://www.wikidata.org/wiki/Q672619) - NIST: http://dlmf.nist.gov/15.2 _a_, _b_, _z_ The [Kummer confluent hypergeometric function](https://en.wikipedia.org/wiki/Confluent_hypergeometric_function), also written $M(a, b, z)$: $$ {}_1F_1(a; b; z) = \sum_{n=0}^{\infty} \frac{(a)_n}{(b)_n} \frac{z^n}{n!} $$ It is an entire function of $z$ and evaluates numerically for any real or complex $z$. ```json example ["Hypergeometric1F1", 1, 2, 2] // ➔ 3.19452804946533 (= (e² − 1)/2) ``` - Wikidata: [Q1331447](https://www.wikidata.org/wiki/Q1331447) - NIST: http://dlmf.nist.gov/13.2 ## Theta and Modular Functions _j_, _z_, _τ_ The [Jacobi theta functions](https://en.wikipedia.org/wiki/Theta_function) $\theta_j(z, \tau)$ for $j \in \{1, 2, 3, 4\}$, with nome $q = e^{i\pi\tau}$ and $\operatorname{Im}(\tau) > 0$: $$ \theta_3(z, \tau) = 1 + 2\sum_{n=1}^{\infty} q^{n^2} \cos(2n\pi z) $$ and similarly for $\theta_1$, $\theta_2$, $\theta_4$. **Convention:** the trigonometric argument is a multiple of $\pi z$ (the functions have period 1 in $z$), matching [Fungrim](https://fungrim.org) and mpmath — not the classical convention with period $\pi$. An optional fourth argument indicates the order of differentiation with respect to $z$; only order 0 currently evaluates numerically. ```json example ["JacobiTheta", 3, 0, "ImaginaryUnit"] // ➔ 1.08643481121331 (= π^(1/4)/Γ(3/4)) ``` - Wikidata: [Q1154532](https://www.wikidata.org/wiki/Q1154532) - NIST: http://dlmf.nist.gov/20.2 _τ_ The [Dedekind eta function](https://en.wikipedia.org/wiki/Dedekind_eta_function), defined on the upper half-plane ($\operatorname{Im}(\tau) > 0$) by: $$ \eta(\tau) = e^{i\pi\tau/12} \prod_{k=1}^{\infty} \left(1 - e^{2\pi i k \tau}\right) $$ It is a modular form of weight $\frac12$, central to the theory of modular functions and integer partitions. ```json example ["DedekindEta", "ImaginaryUnit"] // ➔ 0.768225422326057 (= Γ(1/4)/(2π^(3/4))) ``` - Wikidata: [Q1187208](https://www.wikidata.org/wiki/Q1187208) - NIST: http://dlmf.nist.gov/23.15