SpecialFunctions http://www.openmath.org/CDs/SpecialFunctions 10/Nov/1999 private This content dictionary contains the most commonly used special functions in areas such as applied mathematics, chemistry, science, engineering, physics and statistics. Ying Xue and Stephen Watt, 2000. ExpIntegralE Exponential Integral $E_n(z)$ The exponential integral function is defined by $E_n(z)=\int_{1}^{\infty} \frac{e^{-zt}}{t^n}dt$ where $(n=0, 1, 2, \dots;\ \Re{z}>0)$. It has a branch cut along the negative real axis in the complex $z$ plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [5.1.2] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $E_n(\bar{z})=\overline{E_n(z)}$, symmetry $E_{n+1}(z)=\frac{1}{n}[e^{-z}-zE_n(z)]$, $n=1, 2, \dots$, recurrence relation $\frac{dE_n(z)}{dz}=-E_{n-1}(z)$, $n=1, 2, \dots$, derivative $E_n(0)=\frac{1}{n-1}$, $n>1$, special value $E_0(z)=\frac{e^z}{z}$, special value ExpIntegralEi Exponential Integral $Ei(z)$ The exponential integral function is defined by $Ei(z)=-\int_{-z}^{\infty} \frac{e^{-t}}{t}dt$, where the principal value is taken. It has a branch cut along the negative real axis in the complex $z$ plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [5.1.2] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic $Ei(z)=\gamma+ln(-z)+ \sum_{n=1}^{\infty} \frac{z^n}{n!n}$, series expansion LogIntegral Logarithmic Integral $li(z)$ The logarithmic integral function is defined by $li(z)=\int_{0}^{z} \frac{dt}{lnt}$, where the principal value is taken. It has a branch cut along the negative real axis in the complex $z$ plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [5.1.3] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic $li(z)=Ei(lnz)$ SinIntegral Sine Integral $Si(z)$ The sine integral is defined by $Si(z)=\int_{0}^{z} \frac{sint}{t}dt$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [5.2.1] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic $Si(z)=\sum_{n=0}^{\infty} \frac{(-1)^{n}z^{2n+1}}{(2n+1)(2n+1)!}$, series expansion $Si(-z)=-Si(z)$, symmetry $Si(\bar{z})=\overline{Si(z)}$, symmetry CosIntegral Cosine Integral $Ci(z)$ The cosine integral is defined by $Ci(z)=-\int_z^{\infty} \frac{cost}{t} dt$ where $|argz| \lt \pi$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [5.2.27] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic $Ci(z)=\gamma + lnz + \sum_{n=1}^{\infty} \frac{(-1)^{n}z^{2n}}{2n(2n)!}$, series expansion symmetry relation: $Ci(\bar{z})=\overline{Ci(z)}$, symmetry $Ci(-z)=Ci(z)-i \pi$, ($0 \lt argz \lt \frac{\pi}{2}$), symmetry SinhIntegral Hyperbolic Sine Integral Function $Shi(z)$ The hyperbolic sine integral is defined by $Shi(z)=\int_{0}^{z} \frac{sinht}{t} dt$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [5.2.3] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic $Shi(z)=\sum_{n=0}^{\infty}\frac{z^{2n+1}} {(2n+1)(2n+1)!}$, series expansion CoshIntegral Hyperbolic Cosine Integral Function $Chi(z)$ The hyperbolic cosine integral is defined by $Chi(z)=\gamma + lnz + \int_{0}^{z} \frac{cosht-1}{t} dt$ where $\gamma=.5772156649 \dots$ is Euler's constant and $|argz|\lt \pi$. It has a branch cut along the negative real axis in the complex $z$ plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [5.2.4] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic $Chi(z)=\gamma + lnz + \sum_{n=1}^{\infty} \frac{z^{2n}}{2n(2n)!}$, series expansion Gamma Gamma Function $\Gamma(z)$ The Euler Gamma function is defined by $\Gamma(z)=\int_{0}^{\infty} t^{z-1}e^{-t} dt$. $Gamma(z)$ is a single valued and analytic over the entire complex plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.1.1] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic $\Gamma(n)=1 \cdot 2 \cdot 3 \dots (n-1)n=n!$, integer value $\Gamma(z+1)=z \Gamma(z)=z!$, recurrence relation $\Gamma(n+z)=(n-1+z)(n-2+z)\dots(1+z)\Gamma(1+z)$, recurrence relation $\Gamma(z) \Gamma(1-z)=-z \Gamma(-z) \Gamma(z)$, reflection relation $\Gamma(\frac{1}{2})=(-\frac{1}{2})!$, fractional value $\Gamma(\frac{3}{2})=(\frac{1}{2})!$, fractional value Digamma Digamma Function $\psi(z)$ The Digamma function is defined as $\psi(z)=\frac{d[ln\Gamma(z)]}{dz}=\frac{\Gamma(z)^{\prime}}{\Gamma(z)}$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.3.1] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic $\psi(1)=-\gamma$, integer value $\psi(n)=-\gamma+\sum_{k=1}^{n-1}k^{-1}$, $n \geq 2$, integer value $\psi(z+1)=\psi(z)+\frac{1}{z}$, recurrence relation $\psi(1-z)=\psi(z)+\pi cot\pi z$, reflection relation PolyGamma Polygamma Function $\psi^{(n)}(z)$ The Polygamma function is defined by $\psi^{(n)}(z) =\frac{d^n}{dz^n}\psi(z)=\frac{d^{n+1}}{dz^{n+1}}ln\Gamma(z) =(-1)^{n+1}\int_{0}^{\infty}\frac{t^ne^{-zt}}{1-e^{-t}} dt$ where $n \ge 0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.4.1] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $\psi^{(n)}(1)=(-1)^{n+1}n! \zeta (n+1)$, $n>0$, integer value $\psi^{(n)}(\frac{1}{2})=(-1)^{n+1}n!(2^{n+1}-1) \zeta(n+1)$, $n=1, 2, \dots$, fractional value $\psi^{(n)}(z+1)=\psi^{(n)}(z)+(-1)^nn!z^{-n-1}$, recurrence relation $\psi^{(n)}(z)=(-1)^{n+1}n!\sum_{k=0}^{\infty}(z+k)^{-n-1}$ , $z \neq 0. -1. -2. \dots$, series expansion IncompleteGamma Incomplete Gamma Function $\Gamma(a,z)$ The Incomplete function is defined by the integral $\Gamma(a,z)=\int_{z}^{\infty} e^{-t}t^{a-1}dt$. It has a branch cut along the negative real axis in the complex $z$ plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.5.3] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic Beta Beta Function $B(z,w)$ The Euler beta function is defined as $B(z,w)=\int_{0}^{1} t^{z-1} (1-t)^{w-1} dt$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.2.1] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $B(z, w)=\frac{\Gamma(z)(w)}{\Gamma(z+w)}=B(w, z)$ IncompleteBeta Incomplete Beta Function $B_{z}(a,b)$ The incomplete Euler beta function is defined as $B_{z}(a,b)=\int_{0}^{z} t^{a-1} (1-t)^{b-1} dt$. It has a branch cut along the negative real axis in the complex $z$ plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [6.6.1] Function (real, real, real) $\rightarrow$ real (real, complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $I_{x}(a, b)=B_{x}(a, b)/B(a, b)$ $I_{x}(a, b)=1-I_{1-x}(b, a)$, symmetry $B_{z}(a, b)=a^{-1}z^aF(a, 1-b;a+1;z)$, relation to hypergeometric function erf Error Function $erf(z)$ The error function $erf(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^2}dt$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [7.1.1] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic $erf(-z)=-erf(z)$, symmetry $erf \bar{z}=\overline {erf(z)}$, symmetry erfc Complementary Error Function $erfc(z)$ The complementary error function is defined by $erfc(z)=1-erf(z)$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [7.1.2] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic FresnelC Fresnel Integral Cosine $C(z)$ The Fresnel integral is defined by $C(z)=\int_{0}^{z} cos(\frac{\pi}{2} t^2) dt$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [7.3.1] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic $C(z)=\sum_{n=0}^{\infty} \frac{(-1)^n(\pi /2)^{2n}} {(2n)!(4n+1)}z^{4n+1}$, series expansion $C(-z)=-C(z)$, symmetry $C(iz)=iC(z)$, symmetry $C(\bar{z})=\overline{C(z)}$, symmetry FresnelS Fresnel Integral Sine $S(z)$ The Fresnel integral $S(z)$ is defined by $S(z)=\int_{0}^{z} sin(\frac{\pi}{2}t^2)dt$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [7.3.2] Function real $\rightarrow$ real complex $\rightarrow$ complex symbolic $\rightarrow$ symbolic $S(z)=\sum_{n=0}^{\infty} \frac{(-1)^n(\pi /2)^{2n+1}} {(2n+1)!(4n+3)}z^{4n+3}$, series expansion $S(-z)=-S(z)$, symmetry $S(iz)=-iS(z)$, symmetry $S(\bar{z})=\overline{S(z)}$, symmetry LegendreP Legendre Function of First Kind $P_{\nu}(z)$ The Legendre function $P_{\nu}(z)= F(-\nu, \nu+1;1; \frac{1-z}{2})$ is single-valued analytic function in z-plane excludes z on the real axis from $-\infty$ to $-1$. It satisfies the differential equation $(1-z^2)\frac{d^2w}{dz^2}-2z\frac{dw}{dz}+[\nu(\nu+1) -\frac{\mu^{2}}{1-z^2}]w=0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [8.1.2] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic LegendreQ Legendre Function of Second Kind $Q_{\nu}(z)$ The Legendre function of second kind $Q_{\nu}(z)$. It is analytic single-valued function in z-plane excludes z on the real axis from $-\infty$ to 1. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [8.1.2] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic AssociatedLegendreP Associated Legendre Function of First Kind $P_{\nu}^{\mu}(z)$ The associated Legendre function $P_{\nu}^{\mu}(z)$ is defined by $P_{\nu}^{\mu}(z)=\frac{1}{\Gamma(1-\mu)} (\frac{z+1}{z-1})^{\frac{1}{2}\mu}F(-\nu, \nu+1; 1-\mu; \frac{1-z}{2})$ where $F(a, b; c;z)$ is a hypergeometric function and $|1-z|\lt2$. When $\mu=0$, It reduces to the Legendre function of the first kind $P_{\nu}(z)$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [8.1.2] Function (real, real, real) $\rightarrow$ real (complex, complex, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic) $\rightarrow$ symbolic $P_{-\nu-1}^{\mu}(z)=P_{\nu}^{\mu}(z)$ AssociatedLegendreQ Associated Legendre Function of Second Kind $Q_{\nu}^{\mu}(z)$ The associated Legendre function is defined by $Q_{\nu}^{\mu}(z)=e^{i\mu\pi}2^{-\nu-1}\sqrt{\pi} \frac{\Gamma(\nu+\mu+1)}{\Gamma(\nu+\frac{3}{2})} z^{-\nu-\mu-1} (z^2-1)^{\frac{1}{2}\mu} F(1+\frac{\nu}{2}+\frac{\mu}{2}, \frac{1}{2}+\frac{\nu}{2} ;\nu+\frac{3}{2}; \frac{1}{2})$ where $|z|>1$. When $\mu=0$, It reduces to the Legendre function of the second kind $Q_{\nu}(z)$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [8.1.3] Function (real, real, real) $\rightarrow$ real (complex, complex, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic) $\rightarrow$ symbolic $P_{-\nu-1}^{\mu}(z)=P_{\nu}^{\mu}(z)$ $Q_{-\nu-1}^{\mu}(z)={-\pi e^{i\mu\pi} cos\nu \pi P_{\nu}^{\mu}(z) +Q_{\nu}^{\mu}sin[\pi (\nu+\mu)]}/sin[\pi (\nu-\mu)]$ BesselJ Bessel Functions of Integer Order $J_{\nu}(z)$ The solutions to the differential equation $J_{\pm \nu}(z)$ $z^2 \frac{d^2w}{dz^2}+z\frac{dw}{dz} + (z^2-\nu^2)w=0$ are the Bessel functions of the first kind. It has a branch cut along the negative real axis in the complex $z$ plane. When $\nu=\pm n$, it is analytic in the entire complex z plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.1] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $J_{\nu}(z)=\sum_{k=0}^{\infty} \frac{(-)^k}{k!\Gamma(\nu+k+1)}(\frac{z}{2})^{\nu+2k}$, $|arg z|\lt \pi$, ascending series $J_n^(z)=\frac{1}{\pi} \int_0^{\pi} cos(n \theta- zsin \theta) d \theta$, integral representation $J_{-n}(z)=(-1)^nJ_n(z)$, symmetry $J^{\prime}_0(z)=-J_1(z)$, derivative BesselY Bessel Functions $Y_{\nu}(z)$ $Y_{\nu}(z)$ are the second kind of Bessel functions. They are defined in terms of first kind of Bessel functions as $Y_{\nu}(z)=\frac{J_{\nu}(z)cos(\nu\pi)-J_{-\nu}(z)}{sin(\nu\pi)}$. It has a branch cut along the negative real axis in the complex $z$ plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.1.2] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $Y_{-n}(z)=(-1)^nY_n(z)$, symmetry $Y^{\prime}_0(z)=-Y_1(z)$, derivative HankelH1 Hankel Functions $H_{\nu}^{(1)}(z)$ The Hankel function $H_{\nu}^{(1)}(z)$ is one of the third kind of Bessel function. It is defined by $H_{\nu}^{(1)}(z)=J_{\nu}(z) +iY_{\nu}(z)$. It has a branch cut along the negative real axis in the complex $z$ plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.1.3] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic HankelH2 Hankel Functions $H_{\nu}^{(2)}(z)$ The Hankel function $H_{\nu}^{(2)}(z)$ is one of the third kind of Bessel function. It is defined by $H_{\nu}^{(1)}(z)=J_{\nu}(z)-iY_{\nu}(z)$. It has a branch cut along the negative real axis in the complex $z$ plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.1.4] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic BesselI Modified Bessel Functions $I_{\nu}(z)$ The $I_{\pm\nu}(z)$ are the solutions of the differential equation $z^2\frac{d^2w}{d^2z}+z\frac{dw}{dz}-(z^2+\nu^2)w=0$. They are the modified Bessel functions of the first kind. It has a branch cut along the negative real axis in the complex $z$ plane. When $\nu=\pm n$, $I_{\nu}$ is an entire function of z. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.6.1] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $I_{\nu}(z)=\sum_{k=0}^{\infty} \frac{1}{k!\Gamma(\nu+k+1)} (\frac{z}{2})^{2k+\nu}$, ascending series $I_{-n}(z)=I_n(z)$ BesselK Modified Bessel Functions $K_{\nu}(z)$ The $K_{\nu}(z)$ are the solutions of the differential equation $z^2\frac{d^2w}{d^2z}+z\frac{dw}{dz}-(z^2+\nu^2)w=0$. They are the modified Bessel functions of the second kind. It has a branch cut along the negative real axis in the complex $z$ plane. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.6.1] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $K_{\nu}(z)=\frac{1}{2}\pi \frac{I_{-\nu}(z)-I_{\nu}(z)} {sin(\nu\pi)}$, relation to $I_{\nu}(z)$ $\sqrt{\frac{\pi}{2z}}K_{n+\frac{1}{2}}(z)= \frac{1}{2}\pi(-1)^{n+1} \sqrt{\frac{\pi}{2z}} [I_{n+\frac{1}{2}}(z) -I_{-n-\frac{1}{2}}(z)]$, relation to $I_{\nu}(z)$ $K_{-\nu}(z)=K_{\nu}(z)$ SphericalBesselj Spherical Bessel Functions $j_{n}(z)$ The $j_{n}(z)$ are the spherical Bessel functions of the first kind. It satisfies the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2-n(n+1)]w=0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.1.1] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $j_{n}(z)=\sqrt{\frac{1}{2}\pi/z}J_{n+\frac{1}{2}}(z)$ SphericalBessely Spherical Bessel Functions $y_{n}(z)$ The $y_{n}(z)$ are the spherical Bessel functions of the second kind. It satisfies the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2-n(n+1)]w=0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.1.1] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $y_{n}(z)=\sqrt{\frac{1}{2}\pi/z}Y_{n+\frac{1}{2}}(z)$ SphericalHankel1 Spherical Hankel Functions $h_{n}^{(1)}(z)$ The $h_{n}^{(1)}(z)$ are the spherical Bessel functions of the third kind. It satisfies the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2-n(n+1)]w=0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.1.1] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $h_{n}^{(1)}(z)=\sqrt{\frac{1}{2}\pi/z}H_{n+\frac{1}{2}} ^{(1)}(z)$ SphericalHankel2 Spherical Hankel Functions $h_{n}^{(2)}(z)$ The $h_{n}^{(2)}(z)$ are the spherical Bessel functions of the third kind. It satisfies the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2-n(n+1)]w=0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [9.1] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $h_{n}^{(2)}(z)=\sqrt{\frac{1}{2}\pi/z}H_{n+\frac{1}{2}} ^{(2)}(z)$ ModifiedSphericalBesselI1 Modified Spherical Bessel Functions $I_{n+\frac{1}{2}}(z)$ The $I_{n+\frac{1}{2}}(z)$ are the modified spherical Bessel functions of the first kind. It satisfies the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2+n(n+1)]w=0$ where $n \geq 0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.2.2] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $\sqrt{\frac{1}{2} \pi/z} I_{n+\frac{1}{2}}(z)= e^{-n \pi i/2}j_n(ze^{\pi i/2})$, (-$\pi \le argz \leq \pi/2$) ModifiedSphericalBesselI2 Modified Spherical Bessel Functions $I_{-n-\frac{1}{2}}(z)$ The $I_{-n-\frac{1}{2}}(z)$ are the modified spherical Bessel functions of the second kind. It satisfies the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2+n(n+1)]w=0$ where $n \geq 0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.2.3] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $\sqrt{\frac{1}{2} \pi/z} I_{-n-\frac{1}{2}}(z)= e^{3(n+1) \pi i/2}y_n(ze^{\pi i/2})$, (-$\pi \lt argz \leq \pi/2$) ModifiedSphericalBesselK Modified Spherical Bessel Functions $K_{n+\frac{1}{2}}(z)$ The $K_{n+\frac{1}{2}}(z)$ are the modified spherical Bessel functions of the third kind. It satisfies the differential equation $z^2 w^{\prime\prime}+2zw^{\prime}+[z^2+n(n+1)]w=0$ where $n \geq 0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [10.2.3] Function (integer, real) $\rightarrow$ real (integer, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $\sqrt{\frac{1}{2} \pi/z} K_{n+\frac{1}{2}}(z)= \frac{1}{2} \pi (-1)^{n+1}\sqrt{\frac{1}{2} \pi/z}[I_{n+\frac{1}{2}} -I_{-n-\frac{1}{2}}]$ StruveH Struve Function $H_{\nu}(z)$ The Struve functions $H_{\nu}(z)$ satisfy the differential equation $z^2 w^{\prime\prime}+zw^{\prime}+(z^2-\nu^2)w= \frac{4(\frac{z}{2})^{\nu+1}}{\sqrt{\pi}\Gamma(\nu+\frac{1}{2})}$. The general solution to this equation is $w=aJ_{\nu}(z)+bY_{\nu}(z)+H_{\nu}(z)$, a and b are constants. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [12.1] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $H_{\nu}(z)=(\frac{z}{2})^{\nu+1}\sum_{k=0}^{\infty} \frac{(-1)^k(\frac{1}{2}z)^{2k}}{\Gamma(k+\frac{3}{2}) \Gamma(k+\nu+\frac{3}{2})}$, series expansion StruveL Modified Struve Function $L_{\nu}(z)$ The modified Struve function is defined by $L_{\nu}(z)=-i e^{-\frac{i \nu \pi}{2}}H_{\nu}(iz)$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [12.2] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $L_{\nu}(z)=(\frac{z}{2})^{\nu+1}\sum_{k=0}^{\infty} \frac{(z/2)^{2k}}{\Gamma(k+\frac{3}{2})\Gamma(k+\nu+\frac{3}{2})}$, series expansion AngerJ Anger Function ${\cal{J}}_{\nu}(z)$ The Anger function is defined by ${\cal{J}}_{\nu}(z)=\frac{1}{\pi}\int_{0}^{\pi}cos(\nu\theta-zsin\theta)d\theta$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [12.3.1] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic ${\cal{J}}_{n}(z)=J_{n}(z)$, n integer WeberE Weber Function $E_{\nu}(z)$ The Weber function is defined by $E_{\nu}(z)=\frac{1}{\pi}\int_{0}^{\pi}sin(\nu\theta-zsin\theta)d\theta$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [12.3.3] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $sin(\nu \pi){\cal{J}}_{\nu}(z)=cos(\nu \pi)E_{\nu}(z)-E_{-\nu}(z)$, relation between $E_{\nu}(z)$ and ${\cal{J}}_{\nu}(z)$ $sin(\nu \pi)E_{\nu}(z)={\cal{J}}_{-\nu}(z)-cos(\nu \pi) {\cal{J}}_{\nu}(z)$, relation between $E_{\nu}(z)$ and ${\cal{J}}_{\nu}(z)$ $E_{0}(z)=-H_{0}(z)$, relation between $E_{\nu}(z)$ and $H_{\nu}(z)$ $E_{1}(z)=\frac{2}{\pi}-H_{1}(z)$, relation between $E_{\nu}(z)$ and $H_{\nu}(z)$ $E_{2}(z)=\frac{2z}{3 \pi}-H_{2}(z)$, relation between $E_{\nu}(z)$ and $H_{\nu}(z)$ KummerM Kummer Function $M(a, b, z)$ The Kummer function KummerM satisfies the differential equation $z w^{\prime\prime}+(b-z)w^{\prime}-aw=0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [13.1.1] Function (real, real, real) $\rightarrow$ real (real, real, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic) $\rightarrow$ symbolic $M(a, b, z)=e^z M(b-a, b, -z)$, Kummer transformation KummerU Kummer Function $U(a, b, z)$ The Kummer function KummerU satisfies the differential equation $z w^{\prime\prime}+(b-z)w^{\prime}-aw=0$. Its principal branch is given by $-\pi \lt argz \leq \pi$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [13.1.3] Function (real, real, real) $\rightarrow$ real (real, real, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic) $\rightarrow$ symbolic $U(a, 1-n, z)=z^nU(a+n, 1+n, z)$ $U(a, b, z)=\frac{\pi}{sin \pi b} \{\frac{M(a, b, z)}{\Gamma(1+a-b) \Gamma(b)}-z^{1-b}\frac{M(1+a-b, 2-b, z)}{\Gamma(a)\Gamma(2-b)}\}$, relation between KummerU and KummerM $U(a, b, z)=z^{1-b}U(1+a-b, 2-b, z)$, Kummer transformation WhittakerM Whittaker Function $M_{k,\mu}(z)$ The Whittaker's function WhittakerM satisfies the Whittaker's equation $w^{\prime\prime}+[-\frac{1}{4}+\frac{k}{z}+\frac{(\frac{1}{4} -\mu^2)}{z^2}]w=0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [13.1.32] Function (real, real, real) $\rightarrow$ real (real, real, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic) $\rightarrow$ symbolic Ying: what did you mean here? relation to KummerM WhittakerW Whittaker Function $W_{k,\mu}(z)$ The Whittaker's function WhittakerW satisfies the Whittaker's equation $w^{\prime\prime}+[-\frac{1}{4}+\frac{k}{z}+\frac{(\frac{1}{4} -\mu^2)}{z^2}]w=0$ . M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [13.1.33] Function (real, real, real) $\rightarrow$ real (real, real, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic) $\rightarrow$ symbolic $W_{k, \mu}(z)=e^{-\frac{1}{2}z}z^{\frac{1}{2}+\mu} U(\frac{1}{2}+\mu-k, 1+2\mu, z)$, $-\pi \lt argz \leq \pi$, $k=\frac{1}{2}b-a$, $\nu=\frac{b}{2}-\frac{1}{2}$, relation to KummerU CoulombWaveF Regular Coulomb Wave Function $F_L(\eta,\rho)$ The regular coulomb wave function CoulombWaveF satisfies the Coulomb wave equation $\frac{d^2w}{d^2 \rho}+[1-\frac{\eta}{\rho}-\frac{L(L+1)}{\rho^2}]w=0$ where $\rho \gt0$, $-\infty \lt\eta\lt\infty$ and L is a non-negative integer. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [14.1.3] Function (integer, real, real) $\rightarrow$ real (symbolic, symbolic, symbolic) $\rightarrow$ symbolic $F_L(\eta, \rho)=C_L(\eta)\rho^{L+1}e^{-i\rho}M(L+1-i\eta, 2L+2, 2i\rho)$, relation to KummerM CoulombWaveG Irregular Coulomb Wave Function $G_L(\eta,\rho)$ The irregular Coulomb wave function CoulombWaveG satisfies the Coulomb wave equation $\frac{d^2w}{d^2 \rho}+[1-\frac{\eta}{\rho}-\frac{L(L+1)}{\rho^2}]w=0$ where $\rho \gt0$, $-\infty \lt\eta\lt\infty$ and L is a non-negative integer. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [14.1.14] Function (integer, real, real) $\rightarrow$ real (symbolic, symbolic, symbolic) $\rightarrow$ symbolic $G_L(\eta, \rho)=\frac{2\eta}{C_0^2(\eta)}F_L(\eta, \rho) [ln2\rho+\frac{q_L(\eta)}{p_L(\eta)}]+\theta_L(\eta, \rho)$, $\theta_L(\eta, \rho)=D_L(\eta)\rho^{-L}\psi_L(\eta, \rho)$, $D_L(\eta)C_L(\eta)=\frac{1}{2L+1}$, $\psi_L(\eta, \rho)=\sum_{k=-L}^{\infty} a_k^L(\eta) \rho^{k+L}$ HypergeometricF Hypergeometric Function (Gauss Series) $F(a, b; c; z)$ The hypergeometric function $F(a, b; c; z)$ is the solution of the hypergeometric differential equation $z(1-z)w^{\prime\prime}+[c-(a+b+1)z]w^{\prime}-abw=0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [15.1.1] Function (real, real, real, real) $\rightarrow$ real (complex, complex, complex, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic, symbolic) $\rightarrow$ symbolic $F(1, 1; 2; z)=-z^{-1}ln(1-z)$, special case $F(\frac{1}{2}, 1; \frac{3}{2}; z^2)=\frac{1}{2}z^{-1}ln\frac{(1+z)}{(1-z)}$ $F(\frac{1}{2}, 1; \frac{3}{2}; -z^2)=z^{-}arctanz$, special case theta1 Theta Function $\theta_1(z, q)$ The Jacobi theta function theta1 is defined by $\theta_1(z, q)=2 q^{\frac{1}{4}}\sum_{n=0}^{\infty} (-1)^n q^{n(n+1)} sin(2n+1)z$ where $|q|\lt1$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [16.27.1] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic theta2 Theta Function $\theta_2(z, q)$ The Jacobi theta function theta2 is defined by $\theta_2(z, q)=2 q^{\frac{1}{4}}\sum_{n=0}^{\infty} q^{n(n+1)} cos(2n+1)z$ where $|q|\lt1$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [16.27.2] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic theta3 Theta Function $\theta_3(z, q)$ The Jacobi theta function theta3 is defined by $\theta_3(z, q)=1+2 \sum_{n=1}^{\infty} q^{n^2} cos2nz$ where $|q|\lt1$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [16.27.3] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic theta4 Theta Function $\theta_4(z, q)$ The Jacobi theta function theta4 is defined by $\theta_4(z, q)=1+2 \sum_{n=1}^{\infty} (-1)^n q^{n^2} cos2nz$ where $|q|\lt1$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [16.27.4] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic EllipticF Elliptic Integral of The First Kind $F(\phi\backslash \alpha)$ The elliptic integral of the first kind is defined by $F(\phi\backslash \alpha)=F(\phi|m)=\int_0^{\phi}(1-sin^2\alpha sin^2\theta) ^{-\frac{1}{2}}d\theta$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [17.2.6] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic EllipticE Elliptic Integral of The Second Kind $E(\phi\backslash \alpha)$ The elliptic integral of the second kind is defined by $E(\phi\backslash \alpha)=F(u|m)=\int_0^x(1-t^2)^{-\frac{1}{2}} (1-mt^2)^{\frac{1}{2}}dt$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [17.2.8] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic EllipticPi Elliptic Integral of The Third Kind $\Pi(n; \phi \backslash \alpha)$ The elliptic integral of the third kind is defined by $\Pi(n; \phi \backslash \alpha)=\int_0^{\phi}(1-nsin^2\theta)^{-1} [1-sin^2\alpha sin^2\theta]^{-1/2} d\theta$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [17.2.14] Function (real, real, real) $\rightarrow$ real (complex, complex, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic) $\rightarrow$ symbolic EllipticK Complete Elliptic Integral of The First Kind $K(m)$ The complete elliptic integral of the first kind $[K(m)]=K=\int_0^1[(1-t^2)(1-mt^2)]^{-\frac{1}{2}}dt$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [17.2.6] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic EllipticEK Complete Elliptic Integral of The Second Kind $E[K(m)]$ The complete elliptic integral of the second kind $E[K(m)] = \int_0^1 (1-t^2)^{-\frac{1}{2}} (1-mt^2)^{\frac{1}{2}}dt$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [17.2.6] Function (real, real) $\rightarrow$ real (complex, complex) $\rightarrow$ complex (symbolic, symbolic) $\rightarrow$ symbolic $K=F(\frac{\pi}{2}|m)$ WeierstrassP Weierstrass $\wp$-Function $\wp(z)$ The WeierstrassP($z;g_2, g_3$) is a single-valued doubly periodic function with periods $2w$, $2 w^{\prime}$. $g_2$ and $g_3$ are invariants which are related to $w$ and $w^{\prime}$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [18.1] Function (real, real, real) $\rightarrow$ real (complex, complex, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic) $\rightarrow$ symbolic WeierstrassPPrime Weierstrass $\wp^{\prime}$-Function $\wp^{\prime}(z)$ The WeierstrassPPrime($z;g_2, g_3$) elliptic function is defined by $\wp^{\prime}=\frac{\partial}{\partial{z}} \wp (z; g_2, g_3)$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [18.1] Function (real, real, real) $\rightarrow$ real (complex, complex, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic) $\rightarrow$ symbolic WeierstrassZeta Weierstrass $\zeta$-Function $\zeta(z)$ The WeierstrassZeta($z;g_2, g_3$) function satisfies the differential equation $\zeta^{\prime}(z)=- \wp (z)$. $\zeta(z)$ is not an elliptic function. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [18.1] Function (real, real, real) $\rightarrow$ real (complex, complex, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic) $\rightarrow$ symbolic WeierstrassSigma Weierstrass $\sigma$-Function $\sigma(z)$ The WeierstrassSigma($z;g_2, g_3$) function satisfies the condition $\frac{\sigma^{\prime}(z)}{\sigma(z)}=\zeta(z)$. It is not an elliptic function. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [18.1] Function (real, real, real) $\rightarrow$ real (complex, complex, complex) $\rightarrow$ complex (symbolic, symbolic, symbolic) $\rightarrow$ symbolic MathieuC even Mathieu functions $y(a,q,z)$ The MathieuC functions are solutions to the Mathieu's equation $\frac{d^2y}{dz^2}+(a-2qcos(2z))y=0$. This function is defined to be even in z. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [20.1.1] Function ( real, real, real) $\rightarrow$ real ( symbolic, symbolic, symbolic) $\rightarrow$ symbolic MathieuS odd Mathieu functions $y(a,q,z)$ The MathieuS functions are solutions to the Mathieu's equation $\frac{d^2y}{dz^2}+(a-2qcos(2z))y=0$. This function is defined to be odd in z. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [20.1.1] Function ( real, real, real) $\rightarrow$ real ( symbolic, symbolic, symbolic) $\rightarrow$ symbolic MathieuCharacteristicValuesA characteristic values for even Mathieu functions $a_r(q)$ A countably infinite set of characteristic values $a_r(q)$ yield even periodic solutions to the Mathieu's equation $\frac{d^2y}{dz^2}+(a-2qcos(2z))y=0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [20.1] Function ( real, real, real) $\rightarrow$ real ( symbolic, symbolic, symbolic) $\rightarrow$ symbolic MathieuCharacteristicValuesB characteristic values for odd Mathieu functions $b_r(q)$ A countably infinite set of characteristic values $b_r(q)$ yield odd periodic solutions to the Mathieu's equation $\frac{d^2y}{dz^2}+(a-2qcos(2z))y=0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [20.1] Function ( real, real, real) $\rightarrow$ real ( symbolic, symbolic, symbolic) $\rightarrow$ symbolic JacobiP Jacobi Polynomials $P_n^{(\alpha,\beta)}(x)$ The Jacobi polynomials $P_n^{(\alpha,\beta)}(x)$ satisfy the generating function relation $R^{-1}(1+z+R)^{-\alpha}(1+z+R)^{-\beta} =\sum_{n=0}^{\infty} 2^{-\alpha-\beta} P_n^{(\alpha, \beta)}(x)z^n$ where $R=\sqrt{1-2xz+z^2}$ and $|z|\lt1$. The explicit expression of Jacobi polynomials $P_n^{(\alpha,\beta)}(x)$ is written as $P_n^{(\alpha,\beta)}(x)=\frac{1}{2^n}\sum_{m=0}^n {n+\alpha \choose m}{n+\beta \choose n-m}(x-1)^{n-m}(x+1)^m$ where ($\alpha>-1$, $\beta\gt-1$) . M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [22.9.1], [22.3.1] Function (integer, real, real, real) $\rightarrow$ real (symbolic, symbolic, symbolic, symbolic) $\rightarrow$ symbolic UltrasphericalC Ultraspherical Polynomials $C_n^{(\alpha)}(x)$ The ultraspherical polynomials satisfy the generating function relation $R^{-2 \alpha}=\sum_{n=0}^{\infty} C_n^{(\alpha)}(x) z^n$ where $R=\sqrt{1-2xz+z^2}$ and $|z|\lt1$. The explicit expressions for the Ultraspherical polynomials can be written as $C_n^{(\alpha)}(x)=\frac{1}{\Gamma{(\alpha})} \sum_{m=0}^{[\frac{n}{2}]} (-1)^m \frac{\Gamma(\alpha+n-m)}{m!(n-2m)!} (2x)^{n-2m}$ where $\alpha>-\frac{1}{2}$ and $\alpha\neq 0$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [22.3.4] Function Function (integer, real, real) $\rightarrow$ real (symbolic, symbolic, symbolic) $\rightarrow$ symbolic ChebyshevT Chebyshev Polynomials $T_n(x)$ The Chebyshev polynomials of the first kind $T_n(x)$ satisfy the generating function relation $\frac{1-xz}{R^2}=T_n(x)z^n$ where $|x|\lt1$ and $|z|\lt1$. The explicit Chebyshev polynomials can be written as $T_n(x)=\frac{n}{2} \sum_{m=0}^{[\frac{n}{2}]} (-1)^m \frac{(n-m-1)!}{m!(n-2m)!} (2x)^{n-2m}$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [22.9.9], [22.3.6] Function (integer, real) $\rightarrow$ real (symbolic, symbolic) $\rightarrow$ symbolic ChebyshevU Chebyshev Polynomials $U_n(x)$ The Chebyshev polynomials of the second kind $U_n(x)$ satisfy the generating function relation $R^{-2}=\sum_{n=0}^{\infty} U_n(x)z^n$ where $|x|\lt1$ and $|z|\lt1$. The explicit expression can be written as $U_n(x)=\sum_{m=0}^{[\frac{n}{2}]} (-1)^m \frac{(n-m)!}{m!(n-2m)!} (2x)^{n-2m}$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [22.9.10], [22.3.7] Function (integer, real) $\rightarrow$ real (symbolic, symbolic) $\rightarrow$ symbolic LegendreP Legendre Polynomials $P_n(x)$ The Legendre polynomials satisfy the generating function relation $R^{-1}=\sum_{n=0}^{\infty} P_n(x) z^n$ where $|x|\lt1$ and $|z|\lt1$. The explicit expression for Legendre polynomials can be written as $P_n(x)=\frac{1}{2^n} \sum_{m=0}^{[\frac{n}{2}]} (-1)^m {n \choose m}{2n-2m \choose n} x^{n-2m}$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [22.9.12], [22.3.8] Function (integer, real) $\rightarrow$ real (symbolic, symbolic) $\rightarrow$ symbolic LaguerreL Laguerre Polynomials $L_n^{(\alpha)}(x)$ The Leguerre polynomials $L_n^{(\alpha)}(x)$ satisfy the generating function relation $(1-z)^{-\alpha-1}exp(\frac{xz} {z-1})=\sum_{n=0}^{\infty} L_n^{\alpha}(x) z^n$ where $|z|\lt1$. Its explicit expression can be written as $L_n^{(\alpha)}(x)=\sum_{m=0}^n (-1)^m {n+\alpha \choose n-m} \frac{1}{m!} x^m$ where $\alpha>-1$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [22.9.15], [22.3.9] Function (integer, real, real) $\rightarrow$ real (symbolic, symbolic, symbolic) $\rightarrow$ symbolic HermiteH Hermite Polynomials $H_n(x)$ The Hermite polynomial satisfy the generating function relation $ e^{2xz-z^2}=\sum_{n=0}^{\infty} \frac{1}{n!} H_n(x) z^n$. The explicit expressions for Hermite polynomials can be written as $H_n(x)=n!\sum_{m=0}^{[\frac{n}{2}]} (-1)^m \frac{1}{m!(n-2m)!} (2x)^{n-2m}$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [22.9.17], [22.3.10] Function (integer, real) $\rightarrow$ real (symbolic, symbolic) $\rightarrow$ symbolic Zeta Riemann Zeta Function $\zeta(s)$ The Riemann zeta function is defined by $\zeta(s)=\sum_{k=1}^{\infty} k^{-s}$ where $\Re{s}>1$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [24.2.1] Function real $\rightarrow$ real complex $\rightarrow$ real symbolic $\rightarrow$ symbolic $\zeta(0)=-\frac{1}{2}$, special value $\zeta(1)=\infty$, special value $\zeta^{\prime}(0)=-\frac{1}{2} ln(2 \pi)$, special value $\zeta(-2n)=0$, $n=1,2, \dots$, special value $\zeta(1-2n)=-\frac{B_{2n}}{2n}$, $n=1, 2, \dots$, special value $\zeta(2n)=\frac{(2 \pi)^{2n}}{2(2n)!}|B_{2n}|$, $n=1, 2, \dots$, special value $\zeta(2n+1)=\frac{(-1)^{n+1}(2 \pi)^{2n+1}}{2(2n+1)!} \int_0^1B_{2n+1}(x)cot(\pi x) dx$, $n=1, 2, \dots$, special value $\zeta(2)=1+\frac{1}{2^2}+\frac{1}{3^2}+\dots =\frac{\pi^2}{6}$, special value $\zeta(4)=1+\frac{1}{2^4}+\frac{1}{3^4}+\dots =\frac{\pi^4}{90}$, special value BernoulliB Bernoulli Polynomials $B_n(x)$ The Bernoulli polynomials $B_n(x)$ satisfy the generating function relation $\frac{te^{xt}}{e^t-1}=\sum_{n=0}^{\infty}B_n(x)\frac{t^n}{n!}$ where $|t|\lt2 \pi$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [23.1.1] Function (integer, real) $\rightarrow$ real (symbolic, symbolic) $\rightarrow$ symbolic $B^{\prime}_{n}(x)=nB_{n-1}(x)$, $n=1,2,\dots$, derivative $B_n(x+1)-B_n(x)=nx^{n-1}$, $n=0,1,2,\dots$, difference $B_n(1-x)=(-1)^nB_n(x)$, $n=0,1,2,\dots$, symmetry $(-1)^n B_n(-x)=B_n(x)+nx^{n-1}$, $n=0,1,2,\dots$, symmetry $B_{2n+1}=0$, $n=1, 2, \dots$, special value $B_n(0)=(-1)^n B_n(1)=B_n$, $n=0, 1, \dots$, special value $B_n(\frac{1}{2})=-(1-2^{1-n})B_n$, $n=0, 1, \dots$, special value $B_n(\frac{1}{4})=(-1)^n B_n(\frac{3}{4})$, $n=1, 2, \dots$, special value $B_{2n}(\frac{1}{3})=B_{2n}(\frac{2}{3})$, $n=0, 1, \dots$, special value $B_{2n}(\frac{1}{6})=B_{2n}(\frac{5}{6})$, $n=0, 1, \dots$, special value BernoulliBn Bernoulli Numbers $B_n$ The Bernoulli number $B_n$ is defined as $B_n=B_n(0)$ where $n=0,1,2, \dots$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [23.1.2] Constant $B_0=1$ $B_1=-\frac{1}{2}$ $B_2=\frac{1}{6}$ $B_4=-\frac{1}{30}$ EulerE Euler Polynomials $E_n(x)$ The Euler polynomials $E_n(x)$ satisfy the generating function relation $\frac{2e^{xt}}{e^t+1}=\sum_{n=0}^{\infty}E_n(x) \frac{t^n}{n!}$ where $|t|\lt\pi$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [23.1.1] Function (integer, real) $\rightarrow$ real (symbolic, symbolic) $\rightarrow$ symbolic $E^{\prime}_n(x)=nE_{n-1}(x)$, $n=1,2,\dots$, derivative $E_n(x+1)+E_n(x)=2x^n$, $n=0,1,\dots$, difference $E_n(1-x)=(-1)^nE_n(x)$, $n=0,1,\dots$, symmetry $(-1)^{n+1}E_{n}(-x)=E_{n}(x)-2x^n$, $n=0,1,\dots$, symmetry $E_{2n+1}=0$, $n=0, 1, \dots$, special value $E_n(0)=-E_n(1)$, $n=1, 2, \dots$, special value $E_n(\frac{1}{2})=2^{-n}E_n$, $n=0, 1, \dots$, special value $E_{2n-1}(\frac{1}{3})=-E_{2n-1}(\frac{2}{3})$, $n=1, 2, \dots$, special value EulerE Euler Numbers $E_n$ The Euler number $E_n$ is defined as $E_n=2^nE_n (\frac{1}{2})$ where $n=0,1,2,\dots$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [23.1.2] Constant $E_0=1$ $E_2=-1$ $E_4=5$ Binomial Binomial Coefficients ${n \choose m} $ The definition for the binomial coefficient ${n \choose m} $ is the number of ways of choosing m objects from a collection of n distinct objects without regard to order. It satisfies the generating function relation $(1+x)^n=\sum_{m=0}^n {n \choose m} x^m$ where $n=0,1,2,\dots$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [24.1.1] Function (integer, integer) $\rightarrow$ real (symbolic, symbolic) $\rightarrow$ symbolic ${n \choose m}=\frac{n!}{m!(n-m)!}={n \choose n-m}$, $n \geq m$ ${n+1 \choose m}={n \choose m}+ {n \choose m-1}$, $n\geq m\geq 1$, recurrence relation $\sum_{m=0}^n {r \choose m}{s \choose n-m}= {r+s \choose n}$, $r+s \geq n$, check relation $\sum_{m=0}^n (-1)^{n-m} {r \choose m}={r-1 \choose n}$, $r \geq n+1$, check relation ${n \choose 0}={n \choose 0}=1$, special value ${2n \choose n}=\frac{2^n(2n-1)(2n-3) \dots 3 \cdot 1}{n!}$, special value Multinomial Multinomials $(n;n_1,n_2,\dots, n_m)$ The multinomial coefficients $ (n;n_1,n_2,\dots, n_m)$ satisfy the generating function relation $(x_1+x_2+\dots+x_m)^n=\Sigma(n;n_1,n_2,\dots, n_m) x_1^{n_1}x_2^{n_2}\dots x_m^{n_m}$ summed over $n_1+n_2+\dots+n_m=n$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [24.1.2] Function (integer, ..., integer) $\rightarrow$ real (symbolic, ..., symbolic) $\rightarrow$ symbolic $(n;n_1,n_2,\dots,n_m)=\frac{n!}{n_1!n_2! \dots n_m!}$ $(n+m;n_1+1, n_2+1, \dots, n_m+1)=\sum_{k=1}^m (n+m-1;n_1+1, \dots, n_{k-1}+1, n_{k+1}+1, \dots, n_m+1)$, recurrence relation StirlingS1 Stirling Numbers of the First Kind $S_n^{(m)}$ The Stirling Numbers of the first Kind $S_n^{(m)}$ satisfy the generating function relation $x(x-1)\dots(x-n+1)=\sum_{m=0}^nS_n^{(m)}x^m$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [24.1.3] Function (integer, integer) $\rightarrow$ real (symbolic, symbolic) $\rightarrow$ symbolic $S_{n+1}^{(m)}=S_n^{(m-1)}-nS_n^{(m)}$, $n\geq m\geq1$, recurrence relation ${m \choose r}S_n^{(m)}=\sum_{k=m-r}^{n-r}{n \choose k} S_{n-k}^{(r)}S_k^{(m-r)}$, $n\geq m\geq r$, recurrence relation $\sum_{m=1}^n S_n^{(m)}=0$, $n >1$, check relation $\sum_{m=0}^n (-1)^{n-m} S_n^{(m)}=n!$, check relation $\sum_{k=m}^n S_{n+1}^{(k+1)}n^{k-m}=S_n^{(m)}$, check relation $S_n^{(0)}=\delta_{0n}$, special value $S_n^{(1)}=(-1)^{n-1}(n-1)!$, special value $S_n^{(n-1)}=-{n \choose 2}$, special value $S_n^{(n)}=1$, special value StirlingS2 Stirling Numbers of the Second Kind ${\cal{S}}_n^{(m)}$ The Stirling Numbers of the second kind ${\cal{S}}_n^{(m)}$ satisfy the generating function relation $x^n=\sum_{m=0}^n{\cal{S}}_{n}^{(m)}x(x-1) \dots (x-m+1)$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [24.1.4] Function (integer, integer) $\rightarrow$ real (symbolic, symbolic) $\rightarrow$ symbolic ${\cal{S}}_n^{(m)}=\frac{1}{m!}\sum_{k=0}^m(-1)^{m-k} {m \choose k} k^n$ ${\cal{S}}_{n+1}^{(m)}=m{\cal{S}}_n^{(m)}+{\cal{S}}_n^{(m-1)}$, $n\geq m\geq 1$, recurrence relation ${m \choose r} {\cal{S}}_n^{(m)}=\sum_{k=m-r}^{n-r} {n \choose k} {\cal{S}}_{n-k}^{(r)}{\cal{S}}_k^{(m-r)}$, recurrence relation $\sum_{k=m}^n {\cal{S}}_k^{(m)} {\cal{S}}_n^{(k)}= \sum_{k=m}^n {\cal{S}}_n^{(k)} {\cal{S}}_k^{(m)}=\delta_{mn}$, check relation $\sum_{m=0}^n (-1)^{n-m}m! {\cal{S}}_n^{(m)}=1$, check relation $\sum_{k=m}^n {\cal{S}}_{k-1}^{(m-1)}m^{n-k}={\cal{S}}_n^{(m)}$, check relation ${\cal{S}}_n^{(m)}=\sum_{k=0}^{n-m}(-1)^k {n-1+k \choose n-m+k} {2n-m \choose n-m-k} {\cal{S}}_{n-m+k}^{(k)}$, check relation ${\cal{S}}_n^{(0)}=\delta_{0n}$, special value ${\cal{S}}_n^{(n-1)}={n \choose 2}$, special value ${\cal{S}}_n^{(1)}={\cal{S}}_n^{(n)}=1$, special value PartitionsP Unrestricted Partitions $p(n)$ The number $p(n)$ of unrestricted partitions of the integer n is the number of decompositions of n into integer summands without regard to order. For example, $5=1+4=2+3=1+1+3=1+2+2=1+1+1+2=1+1+1+1+1$ so that $p(5)=7$. $p(n)$ satisfies the generating function relation $\sum_{n=0}^{\infty} p(n)x^n=\Pi_{n=1}^{\infty}(1-x^n)^{-1}$ where $|x|\lt1$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [24.2.1] Function integer $\rightarrow$ real symbolic $\rightarrow$ symbolic $p(n)=\frac{1}{n} \sum_{k=1}^n\sigma_1(k)p(n-k)$, recurrence relation PartitionsQ Unrestricted Partitions $q(n)$ The number $q(n)$ of partitions of the integer n into distinct parts is the number of decompositions of n into distinct integer summands without regard to order. For example, $5=1+4=2+3$ so that $q(5)=3$. It satisfies the generating function relation $\sum_{n=0}^{\infty}q(n)x^n=\Pi_{n=1}^{\infty}(1+x^n)= \Pi_{n=1}^{\infty} (1+x^n)=\Pi_{n=1}^{\infty}(1-x^{2n-1})^{-1}$ where $|x|\lt1$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [24.2.2] Function integer $\rightarrow$ real symbolic $\rightarrow$ symbolic MobiusMu M$\ddot{o}$bius Functions $\mu(n)$ The M$\ddot{o}$bius function $\mu(n)$ is defined to be 1 if $n=1$, $(-1)^k$ if $n$ is the product of k distinct primes and 0 if n contains a square factor. It satisfies the generatingi function relation $\sum_{n=1}^{\infty} \mu(n) n^{-s}=\frac{1}{\zeta(s)}$ for $\Re{s}>1$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [24.3.1] Function integer $\rightarrow$ real symbolic $\rightarrow$ symbolic $\mu(mn)=\mu(m)\mu(n)$ if $(m,n)=1$, recurrence relation $\mu(mn)=0$ if $(m,n)>1$, recurrence relation $\sum_{d \backslash n}\mu(d)=\delta_{n1}$, check relation $\sum_{n=1}^{\infty} \frac{\mu(n)}{n}=0$, asymptotics $\sum_{n=1}^{\infty}\frac{\mu(n)}{n}lnn=-1$, asymptotics EulerTotientPhi Euler Totient Function $\phi(n)$ The Euler Totient $\phi(n)$ function is number of integers not exceeding and relatively prime to n. It satisfies the generating function relation $\sum_{n=1}^{\infty} \phi(n) n^{-s}=\frac{\zeta(s-1)}{\zeta(s)}$ where $\Re{s}>2$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [24.3.2] Function integer $\rightarrow$ real symbolic $\rightarrow$ symbolic $\phi(n)=n\Sigma_{p \backslash n}(1-\frac{1}{p})$ over distinct primes p dividing n. $\phi(mn)=\phi(m) \phi(n)$, $(m,n)=1$, recurrence relation $\Sigma_{d \backslash n}\phi(d)=n$, check relation $\phi(n)=\Sigma_{d \backslash n} \mu (\frac{n}{d})d$, check relation $a^{(\phi(n))}\equiv 1(mod n)$, $(a,n)=1$ DivisorSigma Divisor Function $\sigma_k(n)$ The divisor function $\sigma_k(n)$ is the sum of the $k$-th powers of the divisors of $n$. It satisfies the generating function relation $\sum_{n=1}^{\infty} \sigma_k(n) n^{-s}=\zeta(s)\zeta(s-k)$ where $\Re{s}>k+1$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [24.3.3] Function integer $\rightarrow$ real symbolic $\rightarrow$ symbolic $\sigma_k(mn)=\sigma_k(m) \sigma_k(n)$, $(m,n)=1$, recurrence relation $\sigma_{np}=\sigma_k(n) \sigma_k(p)-p^k \sigma_k(n/p)$, $p$ is prime, recurrence relation Debye Debye Function $f(x)$ The Debye functions are defined by $\int_0^x \frac{t^ndt}{e^t-1}$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [27.1.1] Function (integer, real) $\rightarrow$ real (symbolic, symbolic) $\rightarrow$ symbolic $\int_0^x\frac{t^ndt}{e^t-1}=x^n [\frac{1}{n}- \frac{x}{2(n+1)}+\sum_{k=1}^{\infty}\frac{B_{2k}x^{2k}}{(2k+n)(2k)!}]$ , ($|x|\lt2 \pi$, $n \geq 1$), series representation $\int_0^{\infty}\frac{t^ndt}{e^t-1}=n!\zeta(n+1)$, relation to Riemann Zeta function PlanckRadiationFunc Planck Radiation Function $f(x)$ The Planck radiation function is defined by $f(x)=x^{-5}(e^{1/x}-1)^{-1}$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [27.2] real $\rightarrow$ real symbolic $\rightarrow$ symbolic Function SievertIntegral Sievert Integral $S(x)$ The Sievert Integral is defined by $\int_{0}^{\theta} e^{-xsec\phi} d\phi$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [27.4] Function real $\rightarrow$ real symbolic $\rightarrow$ symbolic $\int_0^{\theta} e^{-xsec \phi}d\phi \sim \sqrt{\frac{\pi}{2}}e^{-x} erf (\sqrt{\frac{x}{2}}\theta)$, ($x \rightarrow \infty$), relation to error function DilogarithmInt Dilogarithm Integral $f(x)$ The Dilogarithm integral is defined by $f(x)=-\int_{1}^{x} \frac{lnt}{t-1}dt$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [27.7] Function real $\rightarrow$ real symbolic $\rightarrow$ symbolic $f(x)=\sum_{k=1}^{\infty} (-1)^k \frac{(x-1)^k}{k^2}$, ($2 \geq x \geq 0$), series expansion ClausenInt Clausen's Integral $f(\theta)$ The Clausen integral is defined as $f(\theta)=-\int_{0}^{\theta}ln(2sin\frac{t}{2})dt =\sum_{k=1}^{\infty}\frac{sink\theta}{k^2}$ $(0 \leq \theta \leq \pi)$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [27.8] Function real $\rightarrow$ real symbolic $\rightarrow$ symbolic $f(\pi-\theta)=f(\theta)-\frac{1}{2}f(2\theta)$, $(0 \leq \theta \leq \frac{\pi}{2})$, functional relationship $\sum_{n=1}^{\infty}\frac{cosn\theta}{n}= -ln(2sin\frac{\theta}{2})$, ($0 \lt \theta \lt 2 \pi$), summable series $\sum_{n=1}^{\infty}\frac{cosn\theta}{n^2}= \frac{\pi^2}{6}-\frac{\pi \theta}{2}+\frac{\theta^2}{4}$, ($0 \leq \theta \leq 2 \pi$), summable series $\sum_{n=1}^{\infty}\frac{cosn\theta}{n^4}= \frac{\pi^4}{90}-\frac{\pi^2 \theta^2}{12}+\frac{\pi \theta^3}{12}- \frac{\theta^4}{48}$, ($0 \leq \theta \leq 2 \pi$), summable series $\sum_{n=1}^{\infty}\frac{sin n \theta}{n}= \frac{1}{2}(\pi-\theta)$, ($0 \lt \theta \lt 2 \pi$), summable series $\sum_{n=1}^{\infty}\frac{sin n \theta}{n^3}= \frac{\pi^2 \theta}{6}-\frac{\pi \theta^2}{4}+\frac{\theta^3}{12}$, ($0 \leq \theta \leq 2 \pi$), summable series $\sum_{n=1}^{\infty}\frac{sin n \theta}{n^5}= \frac{\pi^4 \theta}{90}-\frac{\pi^2 \theta^3}{36}+\frac{\pi \theta^4}{48} -\frac{\theta^5}{240}$, ($0 \leq \theta \leq 2 \pi$), summable series CGCoefficients Clebsch-Gordan Coefficients $(j_1j_2m_1m_2|j_1j_2jm)$ The Clebsch-Gordan Coefficients are defined by $(j_1j_2m_1m_2|j_1j_2jm) = \\ \delta(m, m_1+m_2) \sqrt{\frac{(j_1+j_2-j)!(j+j_1-j_2)!(j+j_2-j_1)!(2j+1)} {(j+j_1+j_2+1)!}} \cdot \Sigma_k \frac{(-1)^k\sqrt{(j_1+m_1)!(j_1-m_1)! (j_2+m_2)!(j_2-m_2)!(j+m)!(j-m)!}}{k!(j_1+j_2-j-k)! (j_1-m_1-k)!(j_2+m_2-k)!(j-j_2+m_1+k)!(j-j_1-m_2+k)!}$ where $\delta(m, m_1+m_2)=1$ when $m=m_1+m_2$ and 0 when $m \neq m_1+m_2$. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, [27.9.1] Function $j_1$, $j_2$, $j=+n$ or $+n/2$ ($n=integer$), condition $j_1+j_2+j=n$, condition $j_1+j_2-j \geq 0$, condition $j_1-j_2+j \geq 0$, condition $-j_1+j_2+j \geq 0$, condition $m_1, m_2, m=\pm n$ or $\pm n/2$, condition $|m_1|\leq j_1$, $|m_2|\leq j_2$, $|m|\leq j$, condition $(j_1j_2m_1m_2|j_1j_2jm)=0$ $m_1+m_2 \neq 0$, condition $(j_10m_10|j_10jm)=\delta(j_1, j) \delta(m_1, m)$, special value $(j_1j_200|j_1j_2j0)=0$ $j_1+j_2+j=2n+1$, special value $(j_1j_1m_1m_1|j_1j_1jm)=0$ $2j_1+j=2n+1$, special value $(j_1j_2m_1m_2|j_1j_2jm)=(-1)^{j_1+j_2-j}(j_1j_2-m_1-m_2|j_1j_2j-m)$, symmetry $(j_1j_2m_1m_2|j_1j_2jm)=(j_2j_1-m_2-m_1|j_2j_1j-m)$, symmetry $(j_1j_2m_1m_2|j_1j_2jm)=(-1)^{j_1+j_2-j} (j_2j_1m_1m_2|j_2j_1jm)$, symmetry $(j_1j_2m_1m_2|j_1j_2jm)=\sqrt{\frac{2j+1}{2j_1+1}} (-1)^{j_2+m_2} (jj_2-mm_2|jj_2j_1-m_1)$, symmetry $(j_1j_2m_1m_2|j_1j_2jm)=\sqrt{\frac{2j+1}{2j_1+1}} (-1)^{j_1-m_1+j-m} (jj_2m-m_2|jj_2j_1m_1)$, symmetry $(j_1j_2m_1m_2|j_1j_2jm)=\sqrt{\frac{2j+1}{2j_1+1}} (-1)^{j-m+j_1-m_1} (j_2jm_2-m|j_2jj_1-m_1)$, symmetry $(j_1j_2m_1m_2|j_1j_2jm)=\sqrt{\frac{2j+1}{2j_2+1}} (-1)^{j_1-m_1} (j_1jm_1-m|j_1jj_2-m_2)$, symmetry $(j_1j_2m_1m_2|j_1j_2jm)=\sqrt{\frac{2j+1}{2j_2+1}} (-1)^{j_1-m_1} (jj_1m-m_1|jj_1j_2m_2)$, symmetry