Bessel and hypergeometric functions

Part of the Fungrim Identities reference — 115 identities for bessel and hypergeometric functions.

Generated reference

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Airy functions (9)
Bessel functions (57)
Confluent hypergeometric functions (11)
Coulomb wave functions (10)
Error functions (12)
Gauss hypergeometric function (16)

Airy functions

$\operatorname{Ai}(z)=z\mathrm{Hypergeometric0F_1}(\frac{4}{3}, \frac{z^3}{9})z\mapsto\operatorname{Ai}(z)^{\prime}(0)+\operatorname{Ai}(0)\mathrm{Hypergeometric0F_1}(\frac{2}{3}, \frac{z^3}{9})$

Holds when $z\in\C$ . Symbols: Hypergeometric0F1 — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. 01bbb6 · Fungrim entry ↗

$z\mapsto\operatorname{Ai}(z)^{\prime}(z)=\frac{1}{2}(\operatorname{Ai}(0)\mathrm{Hypergeometric0F_1}(\frac{5}{3}, \frac{z^3}{9})z^2)+\mathrm{Hypergeometric0F_1}(\frac{1}{3}, \frac{z^3}{9})z\mapsto\operatorname{Ai}(z)^{\prime}(0)$

Holds when $z\in\C$ . Symbols: Hypergeometric0F1 — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. 20e530 · Fungrim entry ↗

$z\mapsto\operatorname{Bi}(z)^{\prime}(z)=\frac{1}{2}(\operatorname{Bi}(0)\mathrm{Hypergeometric0F_1}(\frac{5}{3}, \frac{z^3}{9})z^2)+\mathrm{Hypergeometric0F_1}(\frac{1}{3}, \frac{z^3}{9})z\mapsto\operatorname{Bi}(z)^{\prime}(0)$

Holds when $z\in\C$ . Symbols: Hypergeometric0F1 — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. 4d65e5 · Fungrim entry ↗

$z\mapsto C\operatorname{Ai}(z)+\mathrm{D_{var}}\operatorname{Bi}(z)^{\doubleprime}(z)-z(C\operatorname{Ai}(z)+\mathrm{D_{var}}\operatorname{Bi}(z))=0$

Holds when $z\in\C\land C\in\C\land\mathrm{D_{var}}\in\C$ . Used by the Compute Engine for simplification. 51b241 · Fungrim entry ↗

$z\mapsto\operatorname{Bi}(z)^{\doubleprime}(z)=z\operatorname{Bi}(z)$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. 70ec9f · Fungrim entry ↗

$z\mapsto\operatorname{Ai}(z)^{\doubleprime}(z)=z\operatorname{Ai}(z)$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. b2e9d0 · Fungrim entry ↗

$\operatorname{Bi}(z)=z\mathrm{Hypergeometric0F_1}(\frac{4}{3}, \frac{z^3}{9})z\mapsto\operatorname{Bi}(z)^{\prime}(0)+\operatorname{Bi}(0)\mathrm{Hypergeometric0F_1}(\frac{2}{3}, \frac{z^3}{9})$

Holds when $z\in\C$ . Symbols: Hypergeometric0F1 — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. bd319e · Fungrim entry ↗

$\operatorname{Ai}(z)z\mapsto\operatorname{Bi}(z)^{\prime}(z)-\operatorname{Bi}(z)z\mapsto\operatorname{Ai}(z)^{\prime}(z)=\frac{1}{\pi}$

Used by the Compute Engine for simplification. de9800 · Fungrim entry ↗

$z\mapsto C\operatorname{Ai}(z)+\mathrm{D_{var}}\operatorname{Bi}(z)^{\prime}(z)=(n-2)z\mapsto C\operatorname{Ai}(z)+\mathrm{D_{var}}\operatorname{Bi}(z)^{\prime}(z)+zz\mapsto C\operatorname{Ai}(z)+\mathrm{D_{var}}\operatorname{Bi}(z)^{\prime}(z)$

Holds when $z\in\C\land C\in\C\land\mathrm{D_{var}}\in\C\land n\in3..\infty$ . Used by the Compute Engine for simplification. eadca2 · Fungrim entry ↗

Bessel functions

$\operatorname{K}_{\frac{3}{2}}(z)=(\frac{1}{z}+\frac{1}{z^2})\exp(-z)\sqrt{\frac{\pi z}{2}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 0c09cc · Fungrim entry ↗

$\operatorname{J}_{\frac{1}{2}}(z)=\frac{\sqrt{2}\sin(z)}{\sqrt{\pi}\sqrt{z}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 121b21 · Fungrim entry ↗

$\operatorname{J}_{\nu}(z)=\frac{\sqrt{2}(\mathrm{HypergeometricUStar}(\nu+\frac{1}{2}, 2\nu+1, 2\imaginaryI z)\exp(\imaginaryI((\pi(2\nu+1))/4-z))+\mathrm{HypergeometricUStar}(\nu+\frac{1}{2}, 2\nu+1, -(2\imaginaryI z))\exp(-(\imaginaryI((\pi(2\nu+1))/4-z))))}{2\sqrt{\pi}\sqrt{z}}$

Holds when $0\lt\Re(z)\land\nu\in\C\land z\in\C$ . Symbols: HypergeometricUStar — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. 127f05 · Fungrim entry ↗

$(-n^2+r^2+4r+4)z\mapsto\operatorname{J}_{n}(z)^{\prime}(0)+(r+1)(r+2)z\mapsto\operatorname{J}_{n}(z)^{\prime}(0)=0$

Holds when $n\in\Z\land r\in\N$ . Used by the Compute Engine for simplification. 15ac84 · Fungrim entry ↗

$\operatorname{I}_{n}(z)=\frac{\operatorname{J}_{n}(\imaginaryI z)}{\imaginaryI^{n}}$

Holds when $n\in\Z\land z\in\C$ . Used by the Compute Engine for simplification. 15bbb1 · Fungrim entry ↗

$\mathrm{HankelH_2}(\nu, z)=\operatorname{J}_{\nu}(z)-\imaginaryI\operatorname{Y}_{\nu}(z)$

Holds when $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: HankelH2 — Hankel function of the second kind. Used by the Compute Engine for simplification. 1dce21 · Fungrim entry ↗

$\operatorname{Y}_{\nu}(z)=\frac{\cos(\pi\nu)\operatorname{J}_{\nu}(z)-\operatorname{J}_{-\nu}(z)}{\sin(\pi\nu)}$

Holds when $\nu\in\C\setminus\Z\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 2a4195 · Fungrim entry ↗

$\operatorname{J}_{\nu}(z)=\frac{\sqrt{2}(\mathrm{HypergeometricUStar}(\nu+\frac{1}{2}, 2\nu+1, 2\imaginaryI z)\exp(-(\imaginaryI z))(-(\imaginaryI z))^{-1/2-\nu}+\mathrm{HypergeometricUStar}(\nu+\frac{1}{2}, 2\nu+1, -(2\imaginaryI z))\exp(\imaginaryI z)(\imaginaryI z)^{-1/2-\nu})z^{\nu}}{2\sqrt{\pi}}$

Holds when $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: HypergeometricUStar — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. 32e162 · Fungrim entry ↗

$z\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)=\frac{1}{2}(\operatorname{Y}_{\nu-1}(z)-\operatorname{Y}_{\nu+1}(z))$

Holds when $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 40aeb6 · Fungrim entry ↗

$\operatorname{K}_{\frac{1}{3}}(z)=\frac{\sqrt{3}\pi\operatorname{Ai}(((3z)/2)^{1/3}^2)}{\sqrt[3]{\frac{3}{2}}\sqrt[3]{z}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 49d754 · Fungrim entry ↗

$\operatorname{Y}_{\frac{1}{2}}(z)=-(\frac{\sqrt{2}\cos(z)}{\sqrt{\pi}\sqrt{z}})$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 4dfd41 · Fungrim entry ↗

$\operatorname{I}_{\nu}(z)=\frac{z(\operatorname{I}_{\nu-1}(z)-\operatorname{I}_{\nu+1}(z))}{2\nu}$

Holds when $z\in\C\land\nu\in\Z\setminus\lbrace0\rbrace$ or $\nu\in\C\setminus\lbrace0\rbrace\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 4fb391 · Fungrim entry ↗

$\operatorname{J}_{-n}(z)=\operatorname{J}_{n}(z)\times(-1)^{n}$

Holds when $n\in\Z\land z\in\C$ . Used by the Compute Engine for simplification. 54bce2 · Fungrim entry ↗

$\operatorname{Y}_{\frac{-1}{2}}(z)=\frac{\sqrt{2}\sin(z)}{\sqrt{\pi}\sqrt{z}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 5679f2 · Fungrim entry ↗

$z\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z)=\frac{1}{2}(\operatorname{I}_{\nu-1}(z)+\operatorname{I}_{\nu+1}(z))$

Holds when $\nu\in\Z\land z\in\C$ or $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 58d91f · Fungrim entry ↗

$z\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)=\frac{1}{2}(\operatorname{J}_{\nu-1}(z)-\operatorname{J}_{\nu+1}(z))$

Holds when $\nu\in\Z\land z\in\C$ or $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 5aceb9 · Fungrim entry ↗

$\operatorname{I}_{\frac{-1}{2}}(z)=\frac{\sqrt{2}\cosh(z)}{\sqrt{\pi}\sqrt{z}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 5d9c43 · Fungrim entry ↗

$\operatorname{J}_{\frac{-1}{2}}(z)=\frac{\sqrt{2}\cos(z)}{\sqrt{\pi}\sqrt{z}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 621a9b · Fungrim entry ↗

$(z^2-\nu^2)\operatorname{Y}_{\nu}(z)+z\mapsto\operatorname{Y}_{\nu}(z)^{\doubleprime}(z)z^2+zz\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)=0$

Holds when $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 62f23c · Fungrim entry ↗

$\operatorname{I}_{\frac{3}{2}}(z)=\frac{1}{\sqrt{\pi}}(\sqrt{2}(\frac{\cosh(z)}{z}-\frac{\sinh(z)}{z^2})\sqrt{z})$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 65647f · Fungrim entry ↗

$\operatorname{J}_{\frac{-1}{3}}(z)=\frac{3\operatorname{Ai}(-((3z)/2)^{1/3}^2)+\sqrt{3}\operatorname{Bi}(-((3z)/2)^{1/3}^2)}{2\sqrt[3]{\frac{3}{2}}\sqrt[3]{z}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 685892 · Fungrim entry ↗

$\mathrm{HankelH_1}(\nu, z)=\imaginaryI\operatorname{Y}_{\nu}(z)+\operatorname{J}_{\nu}(z)$

Holds when $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: HankelH1 — Hankel function of the first kind. Used by the Compute Engine for simplification. 6a6a09 · Fungrim entry ↗

$\frac{(r^2+7r+12)z\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z)z^2}{(r+4)!}+\frac{z(2r^2+11r+15)z\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z)}{(r+3)!}+\frac{(-\nu^2-z^2+r(r+4)+4)z\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z)}{(r+2)!}-\frac{2zz\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z)}{(r+1)!}-\frac{1}{r!}(z\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z))=0$

Holds when $\nu\in\C\land r\in\N\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 7377c8 · Fungrim entry ↗

$\operatorname{K}_{\frac{-1}{2}}(z)=\frac{\sqrt{2}\exp(-z)\sqrt{\pi}}{2\sqrt{z}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 7ac286 · Fungrim entry ↗

$\operatorname{I}_{\nu}(z)=\frac{1}{\pi}(\int_{0}^{\pi}\!\cos(\nu t)\exp(z\cos(t))\, \mathrm{d}t)-\frac{1}{\pi}(\sin(\pi\nu)\int_{0}^{\infty}\!\exp(-(\nu t)-z\cosh(t))\, \mathrm{d}t)$

Holds when $0\lt\Re(z)\land\nu\in\C\land z\in\C$ . Used by the Compute Engine for simplification. 7ae3ed · Fungrim entry ↗

$\operatorname{K}_{\nu}(z)=\frac{\sqrt{2}\mathrm{HypergeometricUStar}(\nu+\frac{1}{2}, 2\nu+1, 2z)\exp(-z)\sqrt{\pi}}{2\sqrt{z}}$

$\operatorname{I}_{\nu}(z)=\mathrm{Hypergeometric0F1Regularized}(\nu+1, \frac{z^2}{4})(\frac{z}{2})^{\nu}$

Holds when $\nu\in\N\land z\in\C$ or $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: Hypergeometric0F1Regularized — Regularized confluent hypergeometric limit function. Used by the Compute Engine for simplification. 81eec6 · Fungrim entry ↗

$z\mapsto\operatorname{K}_{0}(z)^{\prime}(z)=-\operatorname{K}_{1}(z)$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 81ffcd · Fungrim entry ↗

$\operatorname{Y}_{\frac{3}{2}}(z)=-(\frac{1}{\sqrt{\pi}}(\sqrt{2}(\frac{\sin(z)}{z}+\frac{\cos(z)}{z^2})\sqrt{z}))$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 8472cc · Fungrim entry ↗

$z\mapsto\operatorname{Y}_{0}(z)^{\prime}(z)=-\operatorname{Y}_{1}(z)$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 8b6264 · Fungrim entry ↗

$-((\nu^2+z^2)\operatorname{I}_{\nu}(z))+z\mapsto\operatorname{I}_{\nu}(z)^{\doubleprime}(z)z^2+zz\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z)=0$

Holds when $\nu\in\Z\land z\in\C$ or $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 95e561 · Fungrim entry ↗

$\operatorname{K}_{\nu}(z)=\frac{\pi(\frac{\mathrm{Hypergeometric0F1Regularized}(1-\nu, z^2/4)}{(z/2)^{\nu}}-\mathrm{Hypergeometric0F1Regularized}(\nu+1, z^2/4)(z/2)^{\nu})}{2\sin(\pi\nu)}$

Holds when $\nu\in\C\setminus\Z\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: Hypergeometric0F1Regularized — Regularized confluent hypergeometric limit function. Used by the Compute Engine for simplification. 98703d · Fungrim entry ↗

$\operatorname{J}_{\nu}(z)=\frac{\mathrm{Hypergeometric1F_1}(\nu+\frac{1}{2}, 2\nu+1, 2\imaginaryI z)\exp(-(\imaginaryI z))(\frac{z}{2})^{\nu}}{\Gamma(\nu+1)}$

Holds when $\nu\in\N\land z\in\C$ or $\nu\notin-\infty..-1\land\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: Hypergeometric1F1 — Kummer confluent hypergeometric function. Used by the Compute Engine for simplification. 9ad254 · Fungrim entry ↗

$\frac{(r^2+7r+12)z\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)z^2}{(r+4)!}+\frac{z(2r^2+11r+15)z\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)}{(r+3)!}+\frac{(-\nu^2+z^2+r(r+4)+4)z\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)}{(r+2)!}+\frac{2zz\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)}{(r+1)!}+\frac{1}{r!}(z\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z))=0$

Holds when $\nu\in\Z\land z\in\C\land r\in\N$ or $\nu\in\C\land r\in\N\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. 9b2f38 · Fungrim entry ↗

$\operatorname{K}_{\nu}(z)=-(\frac{z(\operatorname{K}_{\nu-1}(z)-\operatorname{K}_{\nu+1}(z))}{2\nu})$

$z\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z)=-(\frac{1}{2}(\operatorname{K}_{\nu-1}(z)+\operatorname{K}_{\nu+1}(z)))$

Holds when $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. a0ff0b · Fungrim entry ↗

$\operatorname{J}_{\frac{3}{2}}(z)=\frac{1}{\sqrt{\pi}}(\sqrt{2}(\frac{\sin(z)}{z^2}-\frac{\cos(z)}{z})\sqrt{z})$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. a2a294 · Fungrim entry ↗

$\operatorname{I}_{\frac{1}{2}}(z)=\frac{\sqrt{2}\sinh(z)}{\sqrt{\pi}\sqrt{z}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. a59981 · Fungrim entry ↗

$(z^2-\nu^2)\operatorname{J}_{\nu}(z)+z\mapsto\operatorname{J}_{\nu}(z)^{\doubleprime}(z)z^2+zz\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)=0$

Holds when $\nu\in\Z\land z\in\C$ or $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. ad9caa · Fungrim entry ↗

$\operatorname{I}_{-n}(z)=\operatorname{I}_{n}(z)$

Holds when $n\in\Z\land z\in\C$ . Used by the Compute Engine for simplification. afbd22 · Fungrim entry ↗

$\operatorname{Y}_{\nu}(z)=\frac{\cos(\pi\nu)\mathrm{Hypergeometric0F1Regularized}(\nu+1, -(z^2/4))(z/2)^{\nu}-\frac{\mathrm{Hypergeometric0F1Regularized}(1-\nu, -(z^2/4))}{(z/2)^{\nu}}}{\sin(\pi\nu)}$

$\operatorname{Y}_{\nu}(z)=\frac{z(\operatorname{Y}_{\nu-1}(z)+\operatorname{Y}_{\nu+1}(z))}{2\nu}$

$z\mapsto\operatorname{I}_{0}(z)^{\prime}(z)=\operatorname{I}_{1}(z)$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. c0247f · Fungrim entry ↗

$\operatorname{K}_{\frac{2}{3}}(z)=-(\frac{\sqrt{3}\pi w\mapsto\operatorname{Ai}(w)^{\prime}(((3z)/2)^{1/3}^2)}{3/2^{2/3}z^{2/3}})$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. c362e8 · Fungrim entry ↗

$\operatorname{J}_{\nu}(z)=\frac{1}{\pi}(\int_{0}^{\pi}\!\cos(\nu t-z\sin(t))\, \mathrm{d}t)-\frac{1}{\pi}(\sin(\pi\nu)\int_{0}^{\infty}\!\exp(-(\nu t)-z\sinh(t))\, \mathrm{d}t)$

Holds when $0\lt\Re(z)\land\nu\in\C\land z\in\C$ . Used by the Compute Engine for simplification. cac83e · Fungrim entry ↗

$\operatorname{K}_{\frac{1}{2}}(z)=\frac{\sqrt{2}\exp(-z)\sqrt{\pi}}{2\sqrt{z}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. d1f5c5 · Fungrim entry ↗

$\operatorname{J}_{\frac{1}{3}}(z)=\frac{3\operatorname{Ai}(-((3z)/2)^{1/3}^2)-\sqrt{3}\operatorname{Bi}(-((3z)/2)^{1/3}^2)}{2\sqrt[3]{\frac{3}{2}}\sqrt[3]{z}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. d39c46 · Fungrim entry ↗

$\operatorname{J}_{\nu}(z)=\frac{z(\operatorname{J}_{\nu-1}(z)+\operatorname{J}_{\nu+1}(z))}{2\nu}$

$\operatorname{Y}_{n}(z)=-(\frac{1}{\pi}(2(\operatorname{K}_{n}(\imaginaryI z)\imaginaryI^{n}+(\ln(\imaginaryI z)-\ln(z))\operatorname{J}_{n}(z))))$

Holds when $n\in\Z\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. d5b7e8 · Fungrim entry ↗

$\frac{(r^2+7r+12)z\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z)z^2}{(r+4)!}+\frac{z(2r^2+11r+15)z\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z)}{(r+3)!}+\frac{(-\nu^2-z^2+r(r+4)+4)z\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z)}{(r+2)!}-\frac{2zz\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z)}{(r+1)!}-\frac{1}{r!}(z\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z))=0$

Holds when $\nu\in\Z\land z\in\C\land r\in\N$ or $\nu\in\C\land r\in\N\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. e233b0 · Fungrim entry ↗

$\operatorname{J}_{\frac{2}{3}}(z)=\frac{3w\mapsto\operatorname{Ai}(w)^{\prime}(-((3z)/2)^{1/3}^2)+\sqrt{3}w\mapsto\operatorname{Bi}(w)^{\prime}(-((3z)/2)^{1/3}^2)}{2\frac{3}{2}^{\frac{2}{3}}z^{\frac{2}{3}}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. e72e96 · Fungrim entry ↗

$\frac{(r^2+7r+12)z\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)z^2}{(r+4)!}+\frac{z(2r^2+11r+15)z\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)}{(r+3)!}+\frac{(-\nu^2+z^2+r(r+4)+4)z\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)}{(r+2)!}+\frac{2zz\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)}{(r+1)!}+\frac{1}{r!}(z\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z))=0$

Holds when $\nu\in\C\land r\in\N\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. e85dee · Fungrim entry ↗

$\operatorname{J}_{\nu}(z)=\mathrm{Hypergeometric0F1Regularized}(\nu+1, -(\frac{z^2}{4}))(\frac{z}{2})^{\nu}$

$z\mapsto\operatorname{J}_{0}(z)^{\prime}(z)=-\operatorname{J}_{1}(z)$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. f1afc0 · Fungrim entry ↗

$(-n^2+r^2+4r+4)z\mapsto\operatorname{I}_{n}(z)^{\prime}(0)-(r+1)(r+2)z\mapsto\operatorname{I}_{n}(z)^{\prime}(0)=0$

Holds when $n\in\Z\land r\in\N$ . Used by the Compute Engine for simplification. f303c9 · Fungrim entry ↗

$-((\nu^2+z^2)\operatorname{K}_{\nu}(z))+z\mapsto\operatorname{K}_{\nu}(z)^{\doubleprime}(z)z^2+zz\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z)=0$

Holds when $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. fd9add · Fungrim entry ↗

$\operatorname{K}_{\frac{-1}{3}}(z)=\frac{\sqrt{3}\pi\operatorname{Ai}(((3z)/2)^{1/3}^2)}{\sqrt[3]{\frac{3}{2}}\sqrt[3]{z}}$

Holds when $z\in\C\setminus\lbrace0\rbrace$ . Used by the Compute Engine for simplification. fda595 · Fungrim entry ↗

Confluent hypergeometric functions

$\mathrm{Hypergeometric0F1Regularized}(a, z)=\operatorname{I}_{a-1}(2\sqrt{z})z^{\frac{1-a}{2}}$

Holds when $z\ne0\land a\in\C\land z\in\C$ . Symbols: Hypergeometric0F1Regularized — Regularized confluent hypergeometric limit function. Used by the Compute Engine for simplification. 00dfd1 · Fungrim entry ↗

$\mathrm{Hypergeometric0F_1}(a, z)=\mathrm{Hypergeometric1F_1}(a-\frac{1}{2}, 2a-1, 4\sqrt{z})\exp(-(2\sqrt{z}))$

Holds when $2a\notin-\infty..1\land a\in\C\land z\in\C$ . Symbols: Hypergeometric0F1 — Confluent hypergeometric limit function; Hypergeometric1F1 — Kummer confluent hypergeometric function. Used by the Compute Engine for simplification. 2df3e3 · Fungrim entry ↗

$\mathrm{Hypergeometric0F1Regularized}(a, z)=\operatorname{J}_{a-1}(2\sqrt{-z})(-z)^{\frac{1-a}{2}}$

$\mathrm{HypergeometricUStar}(a, b, z)=\mathrm{Hypergeometric2F_0}(a, a-b+1, -(\frac{1}{z}))$

Holds when $z\ne0\land a\in\C\land b\in\C\land z\in\C$ . Symbols: Hypergeometric2F0 — Tricomi confluent hypergeometric function, alternative notation; HypergeometricUStar — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. 4cf1e9 · Fungrim entry ↗

$\mathrm{HypergeometricU}(a, b, z)=\frac{1}{\Gamma(a)}(\Gamma(b-1)\mathrm{Hypergeometric1F_1}(a-b+1, 2-b, z)z^{1-b})+\frac{\Gamma(1-b)\mathrm{Hypergeometric1F_1}(a, b, z)}{\Gamma(a-b+1)}$

Holds when $z\ne0\land b\notin\Z\land a\in\C\land b\in\C\land z\in\C$ . Symbols: Hypergeometric1F1 — Kummer confluent hypergeometric function; HypergeometricU — Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. 6cf802 · Fungrim entry ↗

$\mathrm{HypergeometricU}(a, b, z)=\mathrm{HypergeometricU}(a-b+1, 2-b, z)z^{1-b}$

Holds when $z\ne0\land a\in\C\land b\in\C\land z\in\C$ . Symbols: HypergeometricU — Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. 9d3147 · Fungrim entry ↗

$\mathrm{Hypergeometric1F1Regularized}(a, b, z)=\mathrm{Hypergeometric1F1Regularized}(b-a, b, -z)\exponentialE^{z}$

Holds when $a\in\C\land b\in\C\land z\in\C$ . Symbols: Hypergeometric1F1Regularized — Regularized Kummer confluent hypergeometric function. Used by the Compute Engine for simplification. a047eb · Fungrim entry ↗

$\mathrm{Hypergeometric1F_1}(a, b, z)=\mathrm{Hypergeometric1F_1}(b-a, b, -z)\exponentialE^{z}$

Holds when $a\in\C\land z\in\C\land b\in\C\setminus\Z_{\le0}$ . Symbols: Hypergeometric1F1 — Kummer confluent hypergeometric function. Used by the Compute Engine for simplification. be533c · Fungrim entry ↗

$\mathrm{HypergeometricUStar}(a, b, z)=\mathrm{HypergeometricU}(a, b, z)z^{a}$

Holds when $z\ne0\land a\in\C\land b\in\C\land z\in\C$ . Symbols: HypergeometricU — Tricomi confluent hypergeometric function; HypergeometricUStar — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. c8fcc7 · Fungrim entry ↗

$\mathrm{HypergeometricUStar}(a, b, z)=\sum_{k=0}^{n-1}\frac{\mathrm{RisingFactorial}(a, k)\mathrm{RisingFactorial}(a-b+1, k)}{k!(-z)^{k}}+\mathrm{HypergeometricUStarRemainder}(n, a, b, z)$

Holds when $z\ne0\land a\in\C\land b\in\C\land z\in\C\land n\in\N$ . Symbols: HypergeometricUStar — Scaled Tricomi confluent hypergeometric function; HypergeometricUStarRemainder — Error term in asymptotic expansion of Tricomi confluent hypergeometric function; RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. d1b3b5 · Fungrim entry ↗

$\mathrm{Hypergeometric1F1Regularized}(a, b, z)=\frac{\frac{\mathrm{HypergeometricUStar}(a, b, z)}{(-z)^{a}}}{\Gamma(b-a)}+\frac{1}{\Gamma(a)}(\mathrm{HypergeometricUStar}(b-a, b, -z)\exponentialE^{z}z^{a-b})$

Holds when $z\ne0\land a\in\C\land b\in\C\land z\in\C$ . Symbols: Hypergeometric1F1Regularized — Regularized Kummer confluent hypergeometric function; HypergeometricUStar — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. f7f84e · Fungrim entry ↗

Coulomb wave functions

$\mathrm{CoulombF}(\ell, \eta, z)=\frac{\mathrm{CoulombH}(1, \ell, \eta, z)-\mathrm{CoulombH}(-1, \ell, \eta, z)}{2\imaginaryI}$

Holds when $\ell+\imaginaryI\eta+1\notin\Z_{\le0}\land\ell-\imaginaryI\eta+1\notin\Z_{\le0}\land\ell\in\C\land\eta\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: CoulombF — Regular Coulomb wave function; CoulombH — Outgoing and ingoing Coulomb wave function. Used by the Compute Engine for simplification. 192a3e · Fungrim entry ↗

$\mathrm{CoulombF}(\ell, \eta, z)=(\frac{\mathrm{HypergeometricUStar}(\ell-\imaginaryI\eta+1, 2\ell+2, 2\imaginaryI z)\exp(-(\imaginaryI z))}{\Gamma(\ell+\imaginaryI\eta+1)(-(2\imaginaryI z))^{\ell-\imaginaryI\eta+1}}+\frac{\mathrm{HypergeometricUStar}(\ell+\imaginaryI\eta+1, 2\ell+2, -(2\imaginaryI z))\exp(\imaginaryI z)}{\Gamma(\ell-\imaginaryI\eta+1)(2\imaginaryI z)^{\ell+\imaginaryI\eta+1}})\times2^{\ell}\exp(\frac{1}{2}(-(\pi\eta)+\mathrm{GammaLn}(\ell+\imaginaryI\eta+1)+\mathrm{GammaLn}(\ell-\imaginaryI\eta+1)))z^{\ell+1}$

Holds when $\ell+\imaginaryI\eta+1\notin\Z_{\le0}\land\ell-\imaginaryI\eta+1\notin\Z_{\le0}\land\ell\in\C\land\eta\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: CoulombF — Regular Coulomb wave function; HypergeometricUStar — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. 1976e1 · Fungrim entry ↗

$z\mapsto\mathrm{CoulombG}(\ell, \eta, z)^{\prime}(z)=(\frac{\eta}{\ell+1}+\frac{\ell+1}{z})\mathrm{CoulombG}(\ell, \eta, z)-\frac{\mathrm{CoulombG}(\ell+1, \eta, z)\sqrt{\ell+\imaginaryI\eta+1}\sqrt{\ell-\imaginaryI\eta+1}}{\ell+1}$

Holds when $\ell\ne-1\land\ell+\imaginaryI\eta+1\notin\Z_{\le0}\land\ell-\imaginaryI\eta+1\notin\Z_{\le0}\land\ell\in\C\land\eta\in\C\land z\in\C\setminus\lparen-\infty, 0\rbrack$ . Symbols: CoulombG — Irregular Coulomb wave function. Used by the Compute Engine for simplification. 2fec14 · Fungrim entry ↗

$\mathrm{CoulombC}(\ell, \eta)=\frac{2^{\ell}\exp(\frac{1}{2}(-(\pi\eta)+\mathrm{GammaLn}(\ell+\imaginaryI\eta+1)+\mathrm{GammaLn}(\ell-\imaginaryI\eta+1)))}{\Gamma(2\ell+2)}$

Holds when $\ell+\imaginaryI\eta+1\notin\Z_{\le0}\land\ell-\imaginaryI\eta+1\notin\Z_{\le0}\land\ell\in\C\land\eta\in\C$ . Symbols: CoulombC — Coulomb wave function Gamow factor. Used by the Compute Engine for simplification. 4a4739 · Fungrim entry ↗

$\mathrm{CoulombG}(\ell, \eta, z)z\mapsto\mathrm{CoulombF}(\ell, \eta, z)^{\prime}(z)-\mathrm{CoulombF}(\ell, \eta, z)z\mapsto\mathrm{CoulombG}(\ell, \eta, z)^{\prime}(z)=1$

Holds when $\ell+\imaginaryI\eta+1\notin\Z_{\le0}\land\ell-\imaginaryI\eta+1\notin\Z_{\le0}\land z\in\C\setminus\lparen-\infty, 0\rbrack$ . Symbols: CoulombF — Regular Coulomb wave function; CoulombG — Irregular Coulomb wave function. Used by the Compute Engine for simplification. 74274a · Fungrim entry ↗

$\mathrm{CoulombG}(\ell, \eta, z)=\frac{1}{2}(\mathrm{CoulombH}(1, \ell, \eta, z)+\mathrm{CoulombH}(-1, \ell, \eta, z))$

Holds when $\ell+\imaginaryI\eta+1\notin\Z_{\le0}\land\ell-\imaginaryI\eta+1\notin\Z_{\le0}\land\ell\in\C\land\eta\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: CoulombG — Irregular Coulomb wave function; CoulombH — Outgoing and ingoing Coulomb wave function. Used by the Compute Engine for simplification. 8547ab · Fungrim entry ↗

$z\mapsto\mathrm{CoulombF}(\ell, \eta, z)^{\prime}(z)=(\frac{\eta}{\ell+1}+\frac{\ell+1}{z})\mathrm{CoulombF}(\ell, \eta, z)-\frac{\mathrm{CoulombF}(\ell+1, \eta, z)\sqrt{\ell+\imaginaryI\eta+1}\sqrt{\ell-\imaginaryI\eta+1}}{\ell+1}$

Holds when $\ell\ne-1\land\ell+\imaginaryI\eta+1\notin\Z_{\le0}\land\ell-\imaginaryI\eta+1\notin\Z_{\le0}\land\ell\in\C\land\eta\in\C\land z\in\C\setminus\lparen-\infty, 0\rbrack$ . Symbols: CoulombF — Regular Coulomb wave function. Used by the Compute Engine for simplification. a51a4b · Fungrim entry ↗

$\mathrm{CoulombG}(\ell, \eta, z)=\frac{\cos(-(\pi(\ell+1/2))-\mathrm{CoulombSigma}((-1)-\ell, \eta)+\mathrm{CoulombSigma}(\ell, \eta))\mathrm{CoulombF}(\ell, \eta, z)-\mathrm{CoulombF}((-1)-\ell, \eta, z)}{\sin(-(\pi(\ell+1/2))-\mathrm{CoulombSigma}((-1)-\ell, \eta)+\mathrm{CoulombSigma}(\ell, \eta))}$

Holds when $2\ell\notin\Z\land\ell+\imaginaryI\eta+1\notin\Z_{\le0}\land\imaginaryI\eta-\ell\notin\Z_{\le0}\land\ell-\imaginaryI\eta+1\notin\Z_{\le0}\land-\ell-\imaginaryI\eta\notin\Z_{\le0}\land\ell\in\C\land\eta\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: CoulombF — Regular Coulomb wave function; CoulombG — Irregular Coulomb wave function; CoulombSigma — Coulomb wave function phase shift. Used by the Compute Engine for simplification. e20938 · Fungrim entry ↗

$\mathrm{CoulombG}(\ell, \eta, z)=\frac{1}{2}(\frac{\mathrm{HypergeometricUStar}(\ell+\imaginaryI\eta+1, 2\ell+2, -(2\imaginaryI z))\exp(\imaginaryI((-\pi\ell)/2+z+\mathrm{CoulombSigma}(\ell, \eta)))}{(2z)^{\imaginaryI\eta}}+\mathrm{HypergeometricUStar}(\ell-\imaginaryI\eta+1, 2\ell+2, 2\imaginaryI z)\exp(-(\imaginaryI((-\pi\ell)/2+z+\mathrm{CoulombSigma}(\ell, \eta))))(2z)^{\imaginaryI\eta})$

Holds when $0\lt\Re(z)\land\ell+\imaginaryI\eta+1\notin\Z_{\le0}\land\ell-\imaginaryI\eta+1\notin\Z_{\le0}\land\ell\in\C\land\eta\in\C\land z\in\C\setminus\lbrace0\rbrace$ . Symbols: CoulombG — Irregular Coulomb wave function; CoulombSigma — Coulomb wave function phase shift; HypergeometricUStar — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. e2efbf · Fungrim entry ↗

$\mathrm{CoulombSigma}(\ell, \eta)=\frac{\mathrm{GammaLn}(\ell+\imaginaryI\eta+1)-\mathrm{GammaLn}(\ell-\imaginaryI\eta+1)}{2\imaginaryI}$

Holds when $\ell+\imaginaryI\eta+1\notin\Z_{\le0}\land\ell-\imaginaryI\eta+1\notin\Z_{\le0}\land\ell\in\C\land\eta\in\C$ . Symbols: CoulombSigma — Coulomb wave function phase shift. Used by the Compute Engine for simplification. ed2bf6 · Fungrim entry ↗

Error functions

$\mathrm{Erfi}(z)=-(\imaginaryI\mathrm{Erf}(\imaginaryI z))$

Holds when $z\in\C$ . Symbols: Erfi — Imaginary error function. Used by the Compute Engine for simplification. 01440f · Fungrim entry ↗

$\mathrm{Erfi}(-z)=-\mathrm{Erfi}(z)$

Holds when $z\in\C$ . Symbols: Erfi — Imaginary error function. Used by the Compute Engine for expansion. 603a49 · Fungrim entry ↗

$\mathrm{Erfc}(z)+\mathrm{Erf}(z)=1$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. 7f355d · Fungrim entry ↗

$\mathrm{Erf}(-z)=-\mathrm{Erf}(z)$

Holds when $z\in\C$ . Used by the Compute Engine for expansion. 94db18 · Fungrim entry ↗

$\mathrm{Erf}(z)=\frac{1}{\sqrt{\pi}}(2z\mathrm{Hypergeometric1F_1}(1, \frac{3}{2}, z^2)\exp(-z^2))$

Holds when $z\in\C$ . Symbols: Hypergeometric1F1 — Kummer confluent hypergeometric function. Used by the Compute Engine for simplification. 98688d · Fungrim entry ↗

$\mathrm{Erf}(z)=\frac{1}{\sqrt{\pi}}(2z\mathrm{Hypergeometric1F_1}(\frac{1}{2}, \frac{3}{2}, -z^2))$

Holds when $z\in\C$ . Symbols: Hypergeometric1F1 — Kummer confluent hypergeometric function. Used by the Compute Engine for simplification. abadc7 · Fungrim entry ↗

$\mathrm{Erfc}(z)=\frac{\mathrm{HypergeometricUStar}(\frac{1}{2}, \frac{1}{2}, z^2)\exp(-z^2)}{z\sqrt{\pi}}$

Holds when $0\lt\Re(z)\land z\in\C$ . Symbols: HypergeometricUStar — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. ae3110 · Fungrim entry ↗

$z\mapsto\mathrm{Erf}(z)^{\prime}(z)=\frac{1}{\sqrt{\pi}}(2\exp(-z^2))$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. b5bd5d · Fungrim entry ↗

$\mathrm{Erfc}(z)=1-\mathrm{Erf}(z)$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. bfc86e · Fungrim entry ↗

$\mathrm{Erf}(z)=\frac{z}{\sqrt{z^2}}-\frac{\mathrm{HypergeometricUStar}(1/2, 1/2, z^2)\exp(-z^2)}{z\sqrt{\pi}}$

Holds when $z\ne0\land z\in\C$ . Symbols: HypergeometricUStar — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. cb93ea · Fungrim entry ↗

$\mathrm{Erfc}(-z)=2-\mathrm{Erfc}(z)$

Holds when $z\in\C$ . Used by the Compute Engine for simplification. ec0205 · Fungrim entry ↗

$z\mapsto\mathrm{Erf}(z)^{\prime}(z)=\frac{1}{\sqrt{\pi}}(2\mathrm{HermitePolynomial}(n-1, z)\times(-1)^{n+1}\exp(-z^2))$

Holds when $z\in\C\land n\in\N^*$ . Used by the Compute Engine for simplification. fae9d3 · Fungrim entry ↗

Gauss hypergeometric function

$\mathrm{Hypergeometric2F_1}(a, b, c, z)=\mathrm{Hypergeometric2F_1}(b, a, c, z)$

Holds when $a\in\C\land b\in\C\land z\in\C\land c\in\C\setminus\Z_{\le0}$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for expansion. 0e0393 · Fungrim entry ↗

$\mathrm{Hypergeometric2F_1}(a, b, c, 0)=1$

Holds when $a\in\C\land b\in\C\land c\in\C\setminus\Z_{\le0}$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 18d955 · Fungrim entry ↗

$\mathrm{Hypergeometric2F_1}(a, b, b, z)=(1-z)^{-a}$

Holds when $a\in\C\land b\in\C\setminus\Z_{\le0}\land z\in\C\setminus\lbrace0, 1\rbrace$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 20bf69 · Fungrim entry ↗

$\frac{1}{\pi}(\sin(\pi(b-a))\mathrm{Hypergeometric2F1Regularized}(a, b, c, z))=\frac{\frac{\mathrm{Hypergeometric2F1Regularized}(a, c-b, a-b+1, 1/(1-z))}{(1-z)^{a}}}{\Gamma(b)\Gamma(c-a)}-\frac{\frac{\mathrm{Hypergeometric2F1Regularized}(b, c-a, -a+b+1, 1/(1-z))}{(1-z)^{b}}}{\Gamma(a)\Gamma(c-b)}$

Holds when $z\notin\lbrack0, \infty\rparen\land a\in\C\land b\in\C\land c\in\C\land z\in\C$ . Symbols: Hypergeometric2F1Regularized — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. 27bc34 · Fungrim entry ↗

$\mathrm{Hypergeometric2F_1}(a, b, c, z)=\mathrm{Hypergeometric2F_1}(a^\star, b^\star, c^\star, z^\star)^\star$

Holds when $a\in\C\land b\in\C\land c\in\C\setminus\Z_{\le0}\land z\in\C\setminus\lbrack1, \infty\rparen$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 3d6d7e · Fungrim entry ↗

$\mathrm{Hypergeometric2F1Regularized}(a, b, c, z)=\frac{\mathrm{Hypergeometric2F1Regularized}(c-a, b, c, \frac{z}{z-1})}{(1-z)^{b}}$

Holds when $z\notin\lbrack1, \infty\rparen\land a\in\C\land b\in\C\land c\in\C\land z\in\C$ . Symbols: Hypergeometric2F1Regularized — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. 504717 · Fungrim entry ↗

$\mathrm{Hypergeometric2F1Regularized}(a, b, c, z)=\mathrm{Hypergeometric2F1Regularized}(c-a, c-b, c, z)(1-z)^{-a-b+c}$

Holds when $z\ne1\land a\in\C\land b\in\C\land c\in\C\land z\in\C$ . Symbols: Hypergeometric2F1Regularized — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. 651a4a · Fungrim entry ↗

$\mathrm{Hypergeometric2F1Regularized}(a, b, -n, z)=\frac{\mathrm{Hypergeometric2F_1}(a+n+1, b+n+1, n+2, z)\mathrm{RisingFactorial}(a, n+1)\mathrm{RisingFactorial}(b, n+1)z^{n+1}}{(n+1)!}$

Holds when $a\in\C\land b\in\C\land n\in\N\land z\in\C\setminus\lbrace1\rbrace$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function; Hypergeometric2F1Regularized — Regularized Gauss hypergeometric function; RisingFactorial — Rising factorial. Used by the Compute Engine for simplification. 65693e · Fungrim entry ↗

$\mathrm{Hypergeometric2F_1}(a, b, c, 1)=\frac{\Gamma(c)\Gamma(-a-b+c)}{\Gamma(c-a)\Gamma(c-b)}$

Holds when $0\lt\Re(-a-b+c)\land a\in\C\land b\in\C\land c\in\C\setminus\Z_{\le0}$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. 659ce8 · Fungrim entry ↗

$\frac{1}{\pi}(\sin(\pi(b-a))\mathrm{Hypergeometric2F1Regularized}(a, b, c, z))=\frac{\frac{\mathrm{Hypergeometric2F1Regularized}(a, a-c+1, a-b+1, 1/z)}{(-z)^{a}}}{\Gamma(b)\Gamma(c-a)}-\frac{\frac{\mathrm{Hypergeometric2F1Regularized}(b, b-c+1, -a+b+1, 1/z)}{(-z)^{b}}}{\Gamma(a)\Gamma(c-b)}$

$\mathrm{Hypergeometric2F_1}(1, 1, 2, z)=-(\frac{\ln(1-z)}{z})$

Holds when $z\in\C\setminus\lbrace0, 1\rbrace$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function. Used by the Compute Engine for simplification. a85994 · Fungrim entry ↗

$\mathrm{Hypergeometric2F1Regularized}(a, b, c, z)=\frac{\mathrm{Hypergeometric2F1Regularized}(a, c-b, c, \frac{z}{z-1})}{(1-z)^{a}}$

$\frac{1}{\pi}(\sin(\pi(-a-b+c))\mathrm{Hypergeometric2F1Regularized}(a, b, c, z))=\frac{\frac{\mathrm{Hypergeometric2F1Regularized}(a, a-c+1, a+b-c+1, 1-1/z)}{z^{a}}}{\Gamma(c-a)\Gamma(c-b)}-\frac{\mathrm{Hypergeometric2F1Regularized}(c-a, 1-a, -a-b+c+1, 1-1/z)z^{a-c}(1-z)^{-a-b+c}}{\Gamma(a)\Gamma(b)}$

Holds when $z\notin\lparen-\infty, 0\rbrack\land z\notin\lbrack1, \infty\rparen\land a\in\C\land b\in\C\land c\in\C\land z\in\C$ . Symbols: Hypergeometric2F1Regularized — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. ca9123 · Fungrim entry ↗

$\frac{1}{\pi}(\sin(\pi(-a-b+c))\mathrm{Hypergeometric2F1Regularized}(a, b, c, z))=\frac{\mathrm{Hypergeometric2F1Regularized}(a, b, a+b-c+1, 1-z)}{\Gamma(c-a)\Gamma(c-b)}-\frac{\mathrm{Hypergeometric2F1Regularized}(c-a, c-b, -a-b+c+1, 1-z)(1-z)^{-a-b+c}}{\Gamma(a)\Gamma(b)}$

$(c-z(a+b+1))z\mapsto\mathrm{Hypergeometric2F_1}(a, b, c, z)^{\prime}(z)-ab\mathrm{Hypergeometric2F_1}(a, b, c, z)+z(1-z)z\mapsto\mathrm{Hypergeometric2F_1}(a, b, c, z)^{\doubleprime}(z)=0$

$\mathrm{Hypergeometric2F1Regularized}(a, b, c, z)=\frac{\mathrm{Hypergeometric2F_1}(a, b, c, z)}{\Gamma(c)}$

Holds when $a\in\C\land b\in\C\land z\in\C\land c\in\C\setminus\Z_{\le0}$ . Symbols: Hypergeometric2F1 — Gauss hypergeometric function; Hypergeometric2F1Regularized — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. fe6e74 · Fungrim entry ↗

Contents​

Airy functions​

Bessel functions​

Confluent hypergeometric functions​

Coulomb wave functions​

Error functions​

Gauss hypergeometric function​

Contents