# Bessel and hypergeometric functions Part of the [Fungrim Identities](/compute-engine/reference/fungrim/) reference — **115 identities** for bessel and hypergeometric functions. :::info[Generated reference] This page is generated from the compiled Fungrim artifact by `scripts/fungrim/gen-reference-doc.ts` (upstream snapshot `3a299164c683`, translator `grim2mathjson 0.1.0`). Do not edit it by hand. The corpus is MIT-licensed; see `data/fungrim/LICENSE`. ::: ## Contents - [Airy functions](#airy-functions) (9) - [Bessel functions](#bessel-functions) (57) - [Confluent hypergeometric functions](#confluent-hypergeometric-functions) (11) - [Coulomb wave functions](#coulomb-wave-functions) (10) - [Error functions](#error-functions) (12) - [Gauss hypergeometric function](#gauss-hypergeometric-function) (16) ## Airy functions $$\operatorname{Ai}(z)=\operatorname{Ai}(0)\mathrm{Hypergeometric0F_1}(\frac{2}{3}, \frac{z^3}{9})+zz\mapsto\operatorname{Ai}(z)^{\prime}(0)\mathrm{Hypergeometric0F_1}(\frac{4}{3}, \frac{z^3}{9})$$ **Holds when** $z\in\C$. **Symbols:** **Hypergeometric0F1** — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. [`01bbb6` · Fungrim entry ↗](https://fungrim.org/entry/01bbb6) --- $$z\mapsto\operatorname{Ai}(z)^{\prime}(z)=z\mapsto\operatorname{Ai}(z)^{\prime}(0)\mathrm{Hypergeometric0F_1}(\frac{1}{3}, \frac{z^3}{9})+\frac{1}{2}(z^2\operatorname{Ai}(0))\mathrm{Hypergeometric0F_1}(\frac{5}{3}, \frac{z^3}{9})$$ **Holds when** $z\in\C$. **Symbols:** **Hypergeometric0F1** — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. [`20e530` · Fungrim entry ↗](https://fungrim.org/entry/20e530) --- $$z\mapsto\operatorname{Bi}(z)^{\prime}(z)=z\mapsto\operatorname{Bi}(z)^{\prime}(0)\mathrm{Hypergeometric0F_1}(\frac{1}{3}, \frac{z^3}{9})+\frac{1}{2}(z^2\operatorname{Bi}(0))\mathrm{Hypergeometric0F_1}(\frac{5}{3}, \frac{z^3}{9})$$ **Holds when** $z\in\C$. **Symbols:** **Hypergeometric0F1** — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. [`4d65e5` · Fungrim entry ↗](https://fungrim.org/entry/4d65e5) --- $$z\mapsto C\operatorname{Ai}(z)+D_{var}\operatorname{Bi}(z)^{\doubleprime}(z)-z(C\operatorname{Ai}(z)+D_{var}\operatorname{Bi}(z))=0$$ **Holds when** $z\in\C\land C\in\C\land D_{var}\in\C$. Used by the Compute Engine for simplification. [`51b241` · Fungrim entry ↗](https://fungrim.org/entry/51b241) --- $$z\mapsto\operatorname{Bi}(z)^{\doubleprime}(z)=z\operatorname{Bi}(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`70ec9f` · Fungrim entry ↗](https://fungrim.org/entry/70ec9f) --- $$z\mapsto\operatorname{Ai}(z)^{\doubleprime}(z)=z\operatorname{Ai}(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`b2e9d0` · Fungrim entry ↗](https://fungrim.org/entry/b2e9d0) --- $$\operatorname{Bi}(z)=\operatorname{Bi}(0)\mathrm{Hypergeometric0F_1}(\frac{2}{3}, \frac{z^3}{9})+zz\mapsto\operatorname{Bi}(z)^{\prime}(0)\mathrm{Hypergeometric0F_1}(\frac{4}{3}, \frac{z^3}{9})$$ **Holds when** $z\in\C$. **Symbols:** **Hypergeometric0F1** — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. [`bd319e` · Fungrim entry ↗](https://fungrim.org/entry/bd319e) --- $$\operatorname{Ai}(z)z\mapsto\operatorname{Bi}(z)^{\prime}(z)-z\mapsto\operatorname{Ai}(z)^{\prime}(z)\operatorname{Bi}(z)=\frac{1}{\pi}$$ Used by the Compute Engine for simplification. [`de9800` · Fungrim entry ↗](https://fungrim.org/entry/de9800) --- $$z\mapsto C\operatorname{Ai}(z)+D_{var}\operatorname{Bi}(z)^{\prime}(z)=zz\mapsto C\operatorname{Ai}(z)+D_{var}\operatorname{Bi}(z)^{\prime}(z)+(n-2)z\mapsto C\operatorname{Ai}(z)+D_{var}\operatorname{Bi}(z)^{\prime}(z)$$ **Holds when** $z\in\C\land n\in3..\infty\land C\in\C\land D_{var}\in\C$. Used by the Compute Engine for simplification. [`eadca2` · Fungrim entry ↗](https://fungrim.org/entry/eadca2) --- ## Bessel functions $$\operatorname{K}_{\frac{3}{2}}(z)=\sqrt{\frac{\pi z}{2}}\exp(-z)(\frac{1}{z}+\frac{1}{z^2})$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`0c09cc` · Fungrim entry ↗](https://fungrim.org/entry/0c09cc) --- $$\operatorname{J}_{\frac{1}{2}}(z)=\frac{1}{z}((\sqrt{2}\sin(z)\sqrt{z})/\sqrt{\pi})$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`121b21` · Fungrim entry ↗](https://fungrim.org/entry/121b21) --- $$\operatorname{J}_{\nu}(z)=\frac{\exp(-(\imaginaryI((\pi(2\nu+1))/4-z)))\mathrm{HypergeometricUStar}(\nu+\frac{1}{2}, 2\nu+1, -(2\imaginaryI z))+\exp(\imaginaryI((\pi(2\nu+1))/4-z))\mathrm{HypergeometricUStar}(\nu+\frac{1}{2}, 2\nu+1, 2\imaginaryI z)}{\sqrt{2\pi z}}$$ **Holds when** $\nu\in\C\land z\in\C\land\Re(z)\gt0$. **Symbols:** **HypergeometricUStar** — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. [`127f05` · Fungrim entry ↗](https://fungrim.org/entry/127f05) --- $$((r^2+4r)-n^2+4)z\mapsto\operatorname{J}_{n}(z)^{\prime}(0)+(r+1)(r+2)z\mapsto\operatorname{J}_{n}(z)^{\prime}(0)=0$$ **Holds when** $n\in\Z\land r\in\N$. Used by the Compute Engine for simplification. [`15ac84` · Fungrim entry ↗](https://fungrim.org/entry/15ac84) --- $$\operatorname{I}_{n}(z)=\frac{\operatorname{J}_{n}(\imaginaryI z)}{\imaginaryI^{n}}$$ **Holds when** $n\in\Z\land z\in\C$. Used by the Compute Engine for simplification. [`15bbb1` · Fungrim entry ↗](https://fungrim.org/entry/15bbb1) --- $$\mathrm{HankelH_2}(\nu, z)=\operatorname{J}_{\nu}(z)-\imaginaryI\operatorname{Y}_{\nu}(z)$$ **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 ↗](https://fungrim.org/entry/1dce21) --- $$\operatorname{Y}_{\nu}(z)=\frac{\cos(\pi\nu)\operatorname{J}_{\nu}(z)-\operatorname{J}_{-\nu}(z)}{\sin(\pi\nu)}$$ **Holds when** $\nu\in\C\setminus\Z\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`2a4195` · Fungrim entry ↗](https://fungrim.org/entry/2a4195) --- $$\operatorname{J}_{\nu}(z)=\frac{\sqrt{2}(\mathrm{HypergeometricUStar}(\nu+\frac{1}{2}, 2\nu+1, 2\imaginaryI z)\exp(-(\imaginaryI z))(-(\imaginaryI z))^{-1/2-\nu}+\mathrm{HypergeometricUStar}(\nu+\frac{1}{2}, 2\nu+1, -(2\imaginaryI z))\exp(\imaginaryI z)(\imaginaryI z)^{-1/2-\nu})z^{\nu}}{2\sqrt{\pi}}$$ **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 ↗](https://fungrim.org/entry/32e162) --- $$z\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)=\frac{1}{2}(\operatorname{Y}_{\nu-1}(z)-\operatorname{Y}_{\nu+1}(z))$$ **Holds when** $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`40aeb6` · Fungrim entry ↗](https://fungrim.org/entry/40aeb6) --- $$\operatorname{K}_{\frac{1}{3}}(z)=\frac{\sqrt{3}\pi\operatorname{Ai}(((3z)/2)^{1/3}^2)}{\sqrt[3]{\frac{3z}{2}}}$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`49d754` · Fungrim entry ↗](https://fungrim.org/entry/49d754) --- $$\operatorname{Y}_{\frac{1}{2}}(z)=-(\frac{1}{z}((2^{1/2}\cos(z)z^{1/2})/\pi^{1/2}))$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`4dfd41` · Fungrim entry ↗](https://fungrim.org/entry/4dfd41) --- $$\operatorname{I}_{\nu}(z)=\frac{z(\operatorname{I}_{\nu-1}(z)-\operatorname{I}_{\nu+1}(z))}{2\nu}$$ **Holds when** $\nu\in\Z\setminus\lbrace0\rbrace\land z\in\C$  _or_  $\nu\in\C\setminus\lbrace0\rbrace\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`4fb391` · Fungrim entry ↗](https://fungrim.org/entry/4fb391) --- $$\operatorname{J}_{-n}(z)=(-1)^{n}\operatorname{J}_{n}(z)$$ **Holds when** $n\in\Z\land z\in\C$. Used by the Compute Engine for simplification. [`54bce2` · Fungrim entry ↗](https://fungrim.org/entry/54bce2) --- $$\operatorname{Y}_{-(\frac{1}{2})}(z)=\frac{1}{z}((\sqrt{2}\sin(z)\sqrt{z})/\sqrt{\pi})$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`5679f2` · Fungrim entry ↗](https://fungrim.org/entry/5679f2) --- $$z\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z)=\frac{1}{2}(\operatorname{I}_{\nu-1}(z)+\operatorname{I}_{\nu+1}(z))$$ **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 ↗](https://fungrim.org/entry/58d91f) --- $$z\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)=\frac{1}{2}(\operatorname{J}_{\nu-1}(z)-\operatorname{J}_{\nu+1}(z))$$ **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 ↗](https://fungrim.org/entry/5aceb9) --- $$\operatorname{I}_{-(\frac{1}{2})}(z)=\frac{1}{z}((\sqrt{2}\cosh(z)\sqrt{z})/\sqrt{\pi})$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`5d9c43` · Fungrim entry ↗](https://fungrim.org/entry/5d9c43) --- $$\operatorname{J}_{-(\frac{1}{2})}(z)=\frac{1}{z}((\sqrt{2}\cos(z)\sqrt{z})/\sqrt{\pi})$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`621a9b` · Fungrim entry ↗](https://fungrim.org/entry/621a9b) --- $$z^2z\mapsto\operatorname{Y}_{\nu}(z)^{\doubleprime}(z)+zz\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)+(z^2-\nu^2)\operatorname{Y}_{\nu}(z)=0$$ **Holds when** $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`62f23c` · Fungrim entry ↗](https://fungrim.org/entry/62f23c) --- $$\operatorname{I}_{\frac{3}{2}}(z)=\sqrt{\frac{2z}{\pi}}(\frac{\cosh(z)}{z}-\frac{\sinh(z)}{z^2})$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`65647f` · Fungrim entry ↗](https://fungrim.org/entry/65647f) --- $$\operatorname{J}_{-(\frac{1}{3})}(z)=\frac{3\operatorname{Ai}(-((3z)/2)^{1/3}^2)+\sqrt{3}\operatorname{Bi}(-((3z)/2)^{1/3}^2)}{2\sqrt[3]{\frac{3z}{2}}}$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`685892` · Fungrim entry ↗](https://fungrim.org/entry/685892) --- $$\mathrm{HankelH_1}(\nu, z)=\operatorname{J}_{\nu}(z)+\imaginaryI\operatorname{Y}_{\nu}(z)$$ **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 ↗](https://fungrim.org/entry/6a6a09) --- $$(\frac{(r^2+7r+12)z\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z)z^2}{(r+4)!}+\frac{z(2r^2+11r+15)z\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z)}{(r+3)!}+\frac{(-\nu^2-z^2+r(r+4)+4)z\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z)}{(r+2)!})-\frac{2zz\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z)}{(r+1)!}-\frac{1}{r!}(z\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z))=0$$ **Holds when** $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace\land r\in\N$. Used by the Compute Engine for simplification. [`7377c8` · Fungrim entry ↗](https://fungrim.org/entry/7377c8) --- $$\operatorname{K}_{-(\frac{1}{2})}(z)=\frac{\sqrt{2}\exp(-z)\sqrt{\pi z}}{2z}$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`7ac286` · Fungrim entry ↗](https://fungrim.org/entry/7ac286) --- $$\operatorname{I}_{\nu}(z)=\frac{1}{\pi}(\int_{0}^{\pi}\!\cos(\nu t)\exp(z\cos(t))\, \mathrm{d}t)-\frac{1}{\pi}(\sin(\pi\nu)\int_{0}^{\infty}\!\exp(-(\nu t)-z\cosh(t))\, \mathrm{d}t)$$ **Holds when** $\nu\in\C\land z\in\C\land\Re(z)\gt0$. Used by the Compute Engine for simplification. [`7ae3ed` · Fungrim entry ↗](https://fungrim.org/entry/7ae3ed) --- $$\operatorname{K}_{\nu}(z)=\frac{\exp(-z)}{\sqrt{\frac{2z}{\pi}}}\mathrm{HypergeometricUStar}(\nu+\frac{1}{2}, 2\nu+1, 2z)$$ **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. [`7efe21` · Fungrim entry ↗](https://fungrim.org/entry/7efe21) --- $$\operatorname{I}_{\nu}(z)=\frac{z}{2}^{\nu}\mathrm{Hypergeometric0F1Regularized}(\nu+1, \frac{z^2}{4})$$ **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 ↗](https://fungrim.org/entry/81eec6) --- $$z\mapsto\operatorname{K}_{0}(z)^{\prime}(z)=-\operatorname{K}_{1}(z)$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`81ffcd` · Fungrim entry ↗](https://fungrim.org/entry/81ffcd) --- $$\operatorname{Y}_{\frac{3}{2}}(z)=-(\sqrt{\frac{2z}{\pi}}(\frac{\cos(z)}{z^2}+\frac{\sin(z)}{z}))$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`8472cc` · Fungrim entry ↗](https://fungrim.org/entry/8472cc) --- $$z\mapsto\operatorname{Y}_{0}(z)^{\prime}(z)=-\operatorname{Y}_{1}(z)$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`8b6264` · Fungrim entry ↗](https://fungrim.org/entry/8b6264) --- $$(z^2z\mapsto\operatorname{I}_{\nu}(z)^{\doubleprime}(z)+zz\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z))-(z^2+\nu^2)\operatorname{I}_{\nu}(z)=0$$ **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 ↗](https://fungrim.org/entry/95e561) --- $$\operatorname{K}_{\nu}(z)=\frac{\pi(\frac{\mathrm{Hypergeometric0F1Regularized}(1-\nu, z^2/4)}{(z/2)^{\nu}}-\mathrm{Hypergeometric0F1Regularized}(\nu+1, z^2/4)(z/2)^{\nu})}{2\sin(\pi\nu)}$$ **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 ↗](https://fungrim.org/entry/98703d) --- $$\operatorname{J}_{\nu}(z)=\frac{\mathrm{Hypergeometric1F_1}(\nu+\frac{1}{2}, 2\nu+1, 2\imaginaryI z)\exp(-(\imaginaryI z))(\frac{z}{2})^{\nu}}{\Gamma(\nu+1)}$$ **Holds when** $\nu\in\N\land z\in\C$  _or_  $\nu\in\C\land\nu\notin-\infty..-1\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`9ad254` · Fungrim entry ↗](https://fungrim.org/entry/9ad254) --- $$\frac{(r^2+7r+12)z\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)z^2}{(r+4)!}+\frac{z(2r^2+11r+15)z\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)}{(r+3)!}+\frac{(-\nu^2+z^2+r(r+4)+4)z\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)}{(r+2)!}+\frac{2zz\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)}{(r+1)!}+\frac{1}{r!}(z\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z))=0$$ **Holds when** $\nu\in\Z\land z\in\C\land r\in\N$  _or_  $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace\land r\in\N$. Used by the Compute Engine for simplification. [`9b2f38` · Fungrim entry ↗](https://fungrim.org/entry/9b2f38) --- $$\operatorname{K}_{\nu}(z)=-(\frac{z(\operatorname{K}_{\nu-1}(z)-\operatorname{K}_{\nu+1}(z))}{2\nu})$$ **Holds when** $\nu\in\Z\setminus\lbrace0\rbrace\land z\in\C$  _or_  $\nu\in\C\setminus\lbrace0\rbrace\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`9d98f8` · Fungrim entry ↗](https://fungrim.org/entry/9d98f8) --- $$z\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z)=-(\frac{1}{2}(\operatorname{K}_{\nu-1}(z)+\operatorname{K}_{\nu+1}(z)))$$ **Holds when** $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`a0ff0b` · Fungrim entry ↗](https://fungrim.org/entry/a0ff0b) --- $$\operatorname{J}_{\frac{3}{2}}(z)=\sqrt{\frac{2z}{\pi}}(\frac{\sin(z)}{z^2}-\frac{\cos(z)}{z})$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`a2a294` · Fungrim entry ↗](https://fungrim.org/entry/a2a294) --- $$\operatorname{I}_{\frac{1}{2}}(z)=\frac{1}{z}((\sqrt{2}\sinh(z)\sqrt{z})/\sqrt{\pi})$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`a59981` · Fungrim entry ↗](https://fungrim.org/entry/a59981) --- $$z^2z\mapsto\operatorname{J}_{\nu}(z)^{\doubleprime}(z)+zz\mapsto\operatorname{J}_{\nu}(z)^{\prime}(z)+(z^2-\nu^2)\operatorname{J}_{\nu}(z)=0$$ **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 ↗](https://fungrim.org/entry/ad9caa) --- $$\operatorname{I}_{-n}(z)=\operatorname{I}_{n}(z)$$ **Holds when** $n\in\Z\land z\in\C$. Used by the Compute Engine for simplification. [`afbd22` · Fungrim entry ↗](https://fungrim.org/entry/afbd22) --- $$\operatorname{Y}_{\nu}(z)=\frac{\cos(\pi\nu)\mathrm{Hypergeometric0F1Regularized}(\nu+1, -(z^2/4))(z/2)^{\nu}-\frac{\mathrm{Hypergeometric0F1Regularized}(1-\nu, -(z^2/4))}{(z/2)^{\nu}}}{\sin(\pi\nu)}$$ **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. [`b049dc` · Fungrim entry ↗](https://fungrim.org/entry/b049dc) --- $$\operatorname{Y}_{\nu}(z)=\frac{z(\operatorname{Y}_{\nu-1}(z)+\operatorname{Y}_{\nu+1}(z))}{2\nu}$$ **Holds when** $\nu\in\Z\setminus\lbrace0\rbrace\land z\in\C$  _or_  $\nu\in\C\setminus\lbrace0\rbrace\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`b6d600` · Fungrim entry ↗](https://fungrim.org/entry/b6d600) --- $$z\mapsto\operatorname{I}_{0}(z)^{\prime}(z)=\operatorname{I}_{1}(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`c0247f` · Fungrim entry ↗](https://fungrim.org/entry/c0247f) --- $$\operatorname{K}_{\frac{2}{3}}(z)=-(\frac{\sqrt{3}\pi w\mapsto\operatorname{Ai}(w)^{\prime}(((3z)/2)^{1/3}^2)}{((3z)/2)^{2/3}})$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`c362e8` · Fungrim entry ↗](https://fungrim.org/entry/c362e8) --- $$\operatorname{J}_{\nu}(z)=\frac{1}{\pi}(\int_{0}^{\pi}\!\cos(\nu t-z\sin(t))\, \mathrm{d}t)-\frac{1}{\pi}(\sin(\pi\nu)\int_{0}^{\infty}\!\exp(-(\nu t)-z\sinh(t))\, \mathrm{d}t)$$ **Holds when** $\nu\in\C\land z\in\C\land\Re(z)\gt0$. Used by the Compute Engine for simplification. [`cac83e` · Fungrim entry ↗](https://fungrim.org/entry/cac83e) --- $$\operatorname{K}_{\frac{1}{2}}(z)=\frac{\sqrt{2}\exp(-z)\sqrt{\pi z}}{2z}$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`d1f5c5` · Fungrim entry ↗](https://fungrim.org/entry/d1f5c5) --- $$\operatorname{J}_{\frac{1}{3}}(z)=\frac{3\operatorname{Ai}(-((3z)/2)^{1/3}^2)-\sqrt{3}\operatorname{Bi}(-((3z)/2)^{1/3}^2)}{2\sqrt[3]{\frac{3z}{2}}}$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`d39c46` · Fungrim entry ↗](https://fungrim.org/entry/d39c46) --- $$\operatorname{J}_{\nu}(z)=\frac{z(\operatorname{J}_{\nu-1}(z)+\operatorname{J}_{\nu+1}(z))}{2\nu}$$ **Holds when** $\nu\in\Z\setminus\lbrace0\rbrace\land z\in\C$  _or_  $\nu\in\C\setminus\lbrace0\rbrace\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`d56914` · Fungrim entry ↗](https://fungrim.org/entry/d56914) --- $$\operatorname{Y}_{n}(z)=\frac{1}{\pi}(-2(\operatorname{K}_{n}(\imaginaryI z)\imaginaryI^{n}+(\ln(\imaginaryI z)-\ln(z))\operatorname{J}_{n}(z)))$$ **Holds when** $n\in\Z\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`d5b7e8` · Fungrim entry ↗](https://fungrim.org/entry/d5b7e8) --- $$(\frac{(r^2+7r+12)z\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z)z^2}{(r+4)!}+\frac{z(2r^2+11r+15)z\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z)}{(r+3)!}+\frac{(-\nu^2-z^2+r(r+4)+4)z\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z)}{(r+2)!})-\frac{2zz\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z)}{(r+1)!}-\frac{1}{r!}(z\mapsto\operatorname{I}_{\nu}(z)^{\prime}(z))=0$$ **Holds when** $\nu\in\Z\land z\in\C\land r\in\N$  _or_  $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace\land r\in\N$. Used by the Compute Engine for simplification. [`e233b0` · Fungrim entry ↗](https://fungrim.org/entry/e233b0) --- $$\operatorname{J}_{\frac{2}{3}}(z)=\frac{3w\mapsto\operatorname{Ai}(w)^{\prime}(-((3z)/2)^{1/3}^2)+\sqrt{3}w\mapsto\operatorname{Bi}(w)^{\prime}(-((3z)/2)^{1/3}^2)}{2(\frac{3z}{2})^{\frac{2}{3}}}$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`e72e96` · Fungrim entry ↗](https://fungrim.org/entry/e72e96) --- $$\frac{(r^2+7r+12)z\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)z^2}{(r+4)!}+\frac{z(2r^2+11r+15)z\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)}{(r+3)!}+\frac{(-\nu^2+z^2+r(r+4)+4)z\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)}{(r+2)!}+\frac{2zz\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z)}{(r+1)!}+\frac{1}{r!}(z\mapsto\operatorname{Y}_{\nu}(z)^{\prime}(z))=0$$ **Holds when** $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace\land r\in\N$. Used by the Compute Engine for simplification. [`e85dee` · Fungrim entry ↗](https://fungrim.org/entry/e85dee) --- $$\operatorname{J}_{\nu}(z)=\frac{z}{2}^{\nu}\mathrm{Hypergeometric0F1Regularized}(\nu+1, -(\frac{z^2}{4}))$$ **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. [`ecd36f` · Fungrim entry ↗](https://fungrim.org/entry/ecd36f) --- $$z\mapsto\operatorname{J}_{0}(z)^{\prime}(z)=-\operatorname{J}_{1}(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`f1afc0` · Fungrim entry ↗](https://fungrim.org/entry/f1afc0) --- $$((r^2+4r)-n^2+4)z\mapsto\operatorname{I}_{n}(z)^{\prime}(0)-(r+1)(r+2)z\mapsto\operatorname{I}_{n}(z)^{\prime}(0)=0$$ **Holds when** $n\in\Z\land r\in\N$. Used by the Compute Engine for simplification. [`f303c9` · Fungrim entry ↗](https://fungrim.org/entry/f303c9) --- $$(z^2z\mapsto\operatorname{K}_{\nu}(z)^{\doubleprime}(z)+zz\mapsto\operatorname{K}_{\nu}(z)^{\prime}(z))-(z^2+\nu^2)\operatorname{K}_{\nu}(z)=0$$ **Holds when** $\nu\in\C\land z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`fd9add` · Fungrim entry ↗](https://fungrim.org/entry/fd9add) --- $$\operatorname{K}_{-(\frac{1}{3})}(z)=\frac{\sqrt{3}\pi\operatorname{Ai}(((3z)/2)^{1/3}^2)}{\sqrt[3]{\frac{3z}{2}}}$$ **Holds when** $z\in\C\setminus\lbrace0\rbrace$. Used by the Compute Engine for simplification. [`fda595` · Fungrim entry ↗](https://fungrim.org/entry/fda595) --- ## Confluent hypergeometric functions $$\mathrm{Hypergeometric0F1Regularized}(a, z)=\sqrt{z}^{1-a}\operatorname{I}_{a-1}(2\sqrt{z})$$ **Holds when** $a\in\C\land z\in\C\land z\ne0$. **Symbols:** **Hypergeometric0F1Regularized** — Regularized confluent hypergeometric limit function. Used by the Compute Engine for simplification. [`00dfd1` · Fungrim entry ↗](https://fungrim.org/entry/00dfd1) --- $$\mathrm{Hypergeometric0F_1}(a, z)=\exp(-(2\sqrt{z}))\mathrm{Hypergeometric1F_1}(a-\frac{1}{2}, 2a-1, 4\sqrt{z})$$ **Holds when** $a\in\C\land z\in\C\land2a\notin-\infty..1$. **Symbols:** **Hypergeometric0F1** — Confluent hypergeometric limit function. Used by the Compute Engine for simplification. [`2df3e3` · Fungrim entry ↗](https://fungrim.org/entry/2df3e3) --- $$\mathrm{Hypergeometric0F1Regularized}(a, z)=\sqrt{-z}^{1-a}\operatorname{J}_{a-1}(2\sqrt{-z})$$ **Holds when** $a\in\C\land z\in\C\land z\ne0$. **Symbols:** **Hypergeometric0F1Regularized** — Regularized confluent hypergeometric limit function. Used by the Compute Engine for simplification. [`325a0e` · Fungrim entry ↗](https://fungrim.org/entry/325a0e) --- $$\mathrm{HypergeometricUStar}(a, b, z)=\mathrm{Hypergeometric2F_0}(a, a-b+1, -(\frac{1}{z}))$$ **Holds when** $a\in\C\land b\in\C\land z\in\C\land z\ne0$. **Symbols:** **Hypergeometric2F0** — Tricomi confluent hypergeometric function, alternative notation; **HypergeometricUStar** — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. [`4cf1e9` · Fungrim entry ↗](https://fungrim.org/entry/4cf1e9) --- $$\mathrm{HypergeometricU}(a, b, z)=\frac{\Gamma(1-b)\mathrm{Hypergeometric1F_1}(a, b, z)}{\Gamma(a-b+1)}+\frac{1}{\Gamma(a)}(\Gamma(b-1)\mathrm{Hypergeometric1F_1}(a-b+1, 2-b, z)z^{1-b})$$ **Holds when** $a\in\C\land b\in\C\land z\in\C\land z\ne0\land b\notin\Z$. **Symbols:** **HypergeometricU** — Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. [`6cf802` · Fungrim entry ↗](https://fungrim.org/entry/6cf802) --- $$\mathrm{HypergeometricU}(a, b, z)=z^{1-b}\mathrm{HypergeometricU}((1+a)-b, 2-b, z)$$ **Holds when** $a\in\C\land b\in\C\land z\in\C\land z\ne0$. **Symbols:** **HypergeometricU** — Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. [`9d3147` · Fungrim entry ↗](https://fungrim.org/entry/9d3147) --- $$\mathrm{Hypergeometric1F1Regularized}(a, b, z)=\exponentialE^{z}\mathrm{Hypergeometric1F1Regularized}(b-a, b, -z)$$ **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 ↗](https://fungrim.org/entry/a047eb) --- $$\mathrm{Hypergeometric1F_1}(a, b, z)=\exponentialE^{z}\mathrm{Hypergeometric1F_1}(b-a, b, -z)$$ **Holds when** $a\in\C\land b\in\C\setminus\Z_{\le0}\land z\in\C$. Used by the Compute Engine for simplification. [`be533c` · Fungrim entry ↗](https://fungrim.org/entry/be533c) --- $$\mathrm{HypergeometricUStar}(a, b, z)=z^{a}\mathrm{HypergeometricU}(a, b, z)$$ **Holds when** $a\in\C\land b\in\C\land z\in\C\land z\ne0$. **Symbols:** **HypergeometricU** — Tricomi confluent hypergeometric function; **HypergeometricUStar** — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. [`c8fcc7` · Fungrim entry ↗](https://fungrim.org/entry/c8fcc7) --- $$\mathrm{HypergeometricUStar}(a, b, z)=\sum_{k=0}^{n-1}\frac{\mathrm{RisingFactorial}(a, k)\mathrm{RisingFactorial}(a-b+1, k)}{k!(-z)^{k}}+\mathrm{HypergeometricUStarRemainder}(n, a, b, z)$$ **Holds when** $a\in\C\land b\in\C\land z\in\C\land z\ne0\land n\in\N$. **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 ↗](https://fungrim.org/entry/d1b3b5) --- $$\mathrm{Hypergeometric1F1Regularized}(a, b, z)=\frac{\frac{\mathrm{HypergeometricUStar}(a, b, z)}{(-z)^{a}}}{\Gamma(b-a)}+\frac{1}{\Gamma(a)}(\mathrm{HypergeometricUStar}(b-a, b, -z)\exponentialE^{z}z^{a-b})$$ **Holds when** $a\in\C\land b\in\C\land z\in\C\land z\ne0$. **Symbols:** **Hypergeometric1F1Regularized** — Regularized Kummer confluent hypergeometric function; **HypergeometricUStar** — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. [`f7f84e` · Fungrim entry ↗](https://fungrim.org/entry/f7f84e) --- ## 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** $\ell\in\C\land\eta\in\C\land(1+\ell+\imaginaryI\eta\notin\Z_{\le0}\land(1+\ell)-\imaginaryI\eta\notin\Z_{\le0})\land z\in\C\setminus\lbrace0\rbrace$. **Symbols:** **CoulombF** — Regular Coulomb wave function; **CoulombH** — Outgoing and ingoing Coulomb wave function. Used by the Compute Engine for simplification. [`192a3e` · Fungrim entry ↗](https://fungrim.org/entry/192a3e) --- $$\mathrm{CoulombF}(\ell, \eta, z)=2^{\ell}z^{\ell+1}\exp(\frac{1}{2}((\mathrm{GammaLn}(1+\ell+\imaginaryI\eta)+\mathrm{GammaLn}((1+\ell)-\imaginaryI\eta))-\pi\eta))(\frac{\exp(\imaginaryI z)\mathrm{HypergeometricUStar}(1+\ell+\imaginaryI\eta, 2\ell+2, -(2\imaginaryI z))}{(2\imaginaryI z)^{1+\ell+\imaginaryI\eta}\Gamma((1+\ell)-\imaginaryI\eta)}+\frac{\exp(-\imaginaryI z)\mathrm{HypergeometricUStar}((1+\ell)-\imaginaryI\eta, 2\ell+2, 2\imaginaryI z)}{(-(2\imaginaryI z))^{(1+\ell)-\imaginaryI\eta}\Gamma(1+\ell+\imaginaryI\eta)})$$ **Holds when** $\ell\in\C\land\eta\in\C\land(1+\ell+\imaginaryI\eta\notin\Z_{\le0}\land(1+\ell)-\imaginaryI\eta\notin\Z_{\le0})\land z\in\C\setminus\lbrace0\rbrace$. **Symbols:** **CoulombF** — Regular Coulomb wave function; **HypergeometricUStar** — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. [`1976e1` · Fungrim entry ↗](https://fungrim.org/entry/1976e1) --- $$z\mapsto\mathrm{CoulombG}(\ell, \eta, z)^{\prime}(z)=(\frac{\ell+1}{z}+\frac{\eta}{\ell+1})\mathrm{CoulombG}(\ell, \eta, z)-\frac{\mathrm{CoulombG}(\ell+1, \eta, z)\sqrt{\ell+\imaginaryI\eta+1}\sqrt{\ell-\imaginaryI\eta+1}}{\ell+1}$$ **Holds when** $\ell\in\C\land\ell\ne-1\land\eta\in\C\land(1+\ell+\imaginaryI\eta\notin\Z_{\le0}\land(1+\ell)-\imaginaryI\eta\notin\Z_{\le0})\land z\in\C\setminus\lparen-\infty, 0\rbrack$. **Symbols:** **CoulombG** — Irregular Coulomb wave function. Used by the Compute Engine for simplification. [`2fec14` · Fungrim entry ↗](https://fungrim.org/entry/2fec14) --- $$\mathrm{CoulombC}(\ell, \eta)=\frac{2^{\ell}\exp(\frac{1}{2}(-(\pi\eta)+\mathrm{GammaLn}(\ell+\imaginaryI\eta+1)+\mathrm{GammaLn}(\ell-\imaginaryI\eta+1)))}{\Gamma(2\ell+2)}$$ **Holds when** $\ell\in\C\land\eta\in\C\land(1+\ell+\imaginaryI\eta\notin\Z_{\le0}\land(1+\ell)-\imaginaryI\eta\notin\Z_{\le0})$. **Symbols:** **CoulombC** — Coulomb wave function Gamow factor. Used by the Compute Engine for simplification. [`4a4739` · Fungrim entry ↗](https://fungrim.org/entry/4a4739) --- $$\mathrm{CoulombG}(\ell, \eta, z)z\mapsto\mathrm{CoulombF}(\ell, \eta, z)^{\prime}(z)-z\mapsto\mathrm{CoulombG}(\ell, \eta, z)^{\prime}(z)\mathrm{CoulombF}(\ell, \eta, z)=1$$ **Holds when** $(1+\ell+\imaginaryI\eta\notin\Z_{\le0}\land(1+\ell)-\imaginaryI\eta\notin\Z_{\le0})\land z\in\C\setminus\lparen-\infty, 0\rbrack$. **Symbols:** **CoulombF** — Regular Coulomb wave function; **CoulombG** — Irregular Coulomb wave function. Used by the Compute Engine for simplification. [`74274a` · Fungrim entry ↗](https://fungrim.org/entry/74274a) --- $$\mathrm{CoulombG}(\ell, \eta, z)=\frac{1}{2}(\mathrm{CoulombH}(1, \ell, \eta, z)+\mathrm{CoulombH}(-1, \ell, \eta, z))$$ **Holds when** $\ell\in\C\land\eta\in\C\land(1+\ell+\imaginaryI\eta\notin\Z_{\le0}\land(1+\ell)-\imaginaryI\eta\notin\Z_{\le0})\land z\in\C\setminus\lbrace0\rbrace$. **Symbols:** **CoulombG** — Irregular Coulomb wave function; **CoulombH** — Outgoing and ingoing Coulomb wave function. Used by the Compute Engine for simplification. [`8547ab` · Fungrim entry ↗](https://fungrim.org/entry/8547ab) --- $$z\mapsto\mathrm{CoulombF}(\ell, \eta, z)^{\prime}(z)=(\frac{\ell+1}{z}+\frac{\eta}{\ell+1})\mathrm{CoulombF}(\ell, \eta, z)-\frac{\mathrm{CoulombF}(\ell+1, \eta, z)\sqrt{\ell+\imaginaryI\eta+1}\sqrt{\ell-\imaginaryI\eta+1}}{\ell+1}$$ **Holds when** $\ell\in\C\land\ell\ne-1\land\eta\in\C\land(1+\ell+\imaginaryI\eta\notin\Z_{\le0}\land(1+\ell)-\imaginaryI\eta\notin\Z_{\le0})\land z\in\C\setminus\lparen-\infty, 0\rbrack$. **Symbols:** **CoulombF** — Regular Coulomb wave function. Used by the Compute Engine for simplification. [`a51a4b` · Fungrim entry ↗](https://fungrim.org/entry/a51a4b) --- $$\mathrm{CoulombG}(\ell, \eta, z)=\frac{\mathrm{CoulombF}(\ell, \eta, z)\cos(\mathrm{CoulombSigma}(\ell, \eta)-\mathrm{CoulombSigma}(-\ell-1, \eta)-(\ell+1/2)\pi)-\mathrm{CoulombF}(-\ell-1, \eta, z)}{\sin(\mathrm{CoulombSigma}(\ell, \eta)-\mathrm{CoulombSigma}(-\ell-1, \eta)-(\ell+1/2)\pi)}$$ **Holds when** $\ell\in\C\land\eta\in\C\land2\ell\notin\Z\land1+\ell+\imaginaryI\eta\notin\Z_{\le0}\land(1+\ell)-\imaginaryI\eta\notin\Z_{\le0}\land\imaginaryI\eta-\ell\notin\Z_{\le0}\land-\ell-\imaginaryI\eta\notin\Z_{\le0}\land z\in\C\setminus\lbrace0\rbrace$. **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 ↗](https://fungrim.org/entry/e20938) --- $$\mathrm{CoulombG}(\ell, \eta, z)=\frac{1}{2}(\frac{\exp(\imaginaryI(z-(\ell\pi)/2+\mathrm{CoulombSigma}(\ell, \eta)))}{(2z)^{\imaginaryI\eta}}\mathrm{HypergeometricUStar}(1+\ell+\imaginaryI\eta, 2\ell+2, -(2\imaginaryI z))+(2z)^{\imaginaryI\eta}\exp(-(\imaginaryI(z-(\ell\pi)/2+\mathrm{CoulombSigma}(\ell, \eta))))\mathrm{HypergeometricUStar}((1+\ell)-\imaginaryI\eta, 2\ell+2, 2\imaginaryI z))$$ **Holds when** $\ell\in\C\land\eta\in\C\land(1+\ell+\imaginaryI\eta\notin\Z_{\le0}\land(1+\ell)-\imaginaryI\eta\notin\Z_{\le0})\land z\in\C\setminus\lbrace0\rbrace\land\Re(z)\gt0$. **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 ↗](https://fungrim.org/entry/e2efbf) --- $$\mathrm{CoulombSigma}(\ell, \eta)=\frac{\mathrm{GammaLn}(1+\ell+\imaginaryI\eta)-\mathrm{GammaLn}((1+\ell)-\imaginaryI\eta)}{2\imaginaryI}$$ **Holds when** $\ell\in\C\land\eta\in\C\land(1+\ell+\imaginaryI\eta\notin\Z_{\le0}\land(1+\ell)-\imaginaryI\eta\notin\Z_{\le0})$. **Symbols:** **CoulombSigma** — Coulomb wave function phase shift. Used by the Compute Engine for simplification. [`ed2bf6` · Fungrim entry ↗](https://fungrim.org/entry/ed2bf6) --- ## Error functions $$\mathrm{Erfi}(z)=-(\imaginaryI\mathrm{Erf}(\imaginaryI z))$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`01440f` · Fungrim entry ↗](https://fungrim.org/entry/01440f) --- $$\mathrm{Erfi}(-z)=-\mathrm{Erfi}(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`603a49` · Fungrim entry ↗](https://fungrim.org/entry/603a49) --- $$\mathrm{Erf}(z)+\mathrm{Erfc}(z)=1$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`7f355d` · Fungrim entry ↗](https://fungrim.org/entry/7f355d) --- $$\mathrm{Erf}(-z)=-\mathrm{Erf}(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for expansion. [`94db18` · Fungrim entry ↗](https://fungrim.org/entry/94db18) --- $$\mathrm{Erf}(z)=(2z\mathrm{Hypergeometric1F_1}(1, \frac{3}{2}, z^2)\exp(-z^2))/\sqrt{\pi}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`98688d` · Fungrim entry ↗](https://fungrim.org/entry/98688d) --- $$\mathrm{Erf}(z)=(2z\mathrm{Hypergeometric1F_1}(\frac{1}{2}, \frac{3}{2}, -z^2))/\sqrt{\pi}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`abadc7` · Fungrim entry ↗](https://fungrim.org/entry/abadc7) --- $$\mathrm{Erfc}(z)=\frac{\mathrm{HypergeometricUStar}(\frac{1}{2}, \frac{1}{2}, z^2)\exp(-z^2)}{z\sqrt{\pi}}$$ **Holds when** $z\in\C\land\Re(z)\gt0$. **Symbols:** **HypergeometricUStar** — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. [`ae3110` · Fungrim entry ↗](https://fungrim.org/entry/ae3110) --- $$z\mapsto\mathrm{Erf}(z)^{\prime}(z)=(2\exp(-z^2))/\sqrt{\pi}$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`b5bd5d` · Fungrim entry ↗](https://fungrim.org/entry/b5bd5d) --- $$\mathrm{Erfc}(z)=1-\mathrm{Erf}(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`bfc86e` · Fungrim entry ↗](https://fungrim.org/entry/bfc86e) --- $$\mathrm{Erf}(z)=\frac{z}{\sqrt{z^2}}-\frac{\mathrm{HypergeometricUStar}(1/2, 1/2, z^2)\exp(-z^2)}{z\sqrt{\pi}}$$ **Holds when** $z\in\C\land z\ne0$. **Symbols:** **HypergeometricUStar** — Scaled Tricomi confluent hypergeometric function. Used by the Compute Engine for simplification. [`cb93ea` · Fungrim entry ↗](https://fungrim.org/entry/cb93ea) --- $$\mathrm{Erfc}(-z)=2-\mathrm{Erfc}(z)$$ **Holds when** $z\in\C$. Used by the Compute Engine for simplification. [`ec0205` · Fungrim entry ↗](https://fungrim.org/entry/ec0205) --- $$z\mapsto\mathrm{Erf}(z)^{\prime}(z)=(2\mathrm{HermitePolynomial}(n-1, z)\times(-1)^{n+1}\exp(-z^2))/\sqrt{\pi}$$ **Holds when** $z\in\C\land n\in\N^*$. Used by the Compute Engine for simplification. [`fae9d3` · Fungrim entry ↗](https://fungrim.org/entry/fae9d3) --- ## Gauss hypergeometric function $$\mathrm{Hypergeometric2F_1}(a, b, c, z)=\mathrm{Hypergeometric2F_1}(b, a, c, z)$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\setminus\Z_{\le0}\land z\in\C$. Used by the Compute Engine for expansion. [`0e0393` · Fungrim entry ↗](https://fungrim.org/entry/0e0393) --- $$\mathrm{Hypergeometric2F_1}(a, b, c, 0)=1$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\setminus\Z_{\le0}$. Used by the Compute Engine for simplification. [`18d955` · Fungrim entry ↗](https://fungrim.org/entry/18d955) --- $$\mathrm{Hypergeometric2F_1}(a, b, b, z)=(1-z)^{-a}$$ **Holds when** $a\in\C\land b\in\C\setminus\Z_{\le0}\land z\in\C\setminus\lbrace0, 1\rbrace$. Used by the Compute Engine for simplification. [`20bf69` · Fungrim entry ↗](https://fungrim.org/entry/20bf69) --- $$\frac{1}{\pi}(\sin(\pi(b-a))\mathrm{Hypergeometric2F1Regularized}(a, b, c, z))=\frac{\frac{\mathrm{Hypergeometric2F1Regularized}(a, c-b, a-b+1, 1/(1-z))}{(1-z)^{a}}}{\Gamma(b)\Gamma(c-a)}-\frac{\frac{\mathrm{Hypergeometric2F1Regularized}(b, c-a, -a+b+1, 1/(1-z))}{(1-z)^{b}}}{\Gamma(a)\Gamma(c-b)}$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\land z\in\C\land z\notin\lbrack0, \infty\rparen$. **Symbols:** **Hypergeometric2F1Regularized** — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. [`27bc34` · Fungrim entry ↗](https://fungrim.org/entry/27bc34) --- $$\mathrm{Hypergeometric2F_1}(a, b, c, z)=\mathrm{Hypergeometric2F_1}(a^\star, b^\star, c^\star, z^\star)^\star$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\setminus\Z_{\le0}\land z\in\C\setminus\lbrack1, \infty\rparen$. Used by the Compute Engine for simplification. [`3d6d7e` · Fungrim entry ↗](https://fungrim.org/entry/3d6d7e) --- $$\mathrm{Hypergeometric2F1Regularized}(a, b, c, z)=\frac{\mathrm{Hypergeometric2F1Regularized}(c-a, b, c, \frac{z}{z-1})}{(1-z)^{b}}$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\land z\in\C\land z\notin\lbrack1, \infty\rparen$. **Symbols:** **Hypergeometric2F1Regularized** — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. [`504717` · Fungrim entry ↗](https://fungrim.org/entry/504717) --- $$\mathrm{Hypergeometric2F1Regularized}(a, b, c, z)=(1-z)^{c-a-b}\mathrm{Hypergeometric2F1Regularized}(c-a, c-b, c, z)$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\land z\in\C\land z\ne1$. **Symbols:** **Hypergeometric2F1Regularized** — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. [`651a4a` · Fungrim entry ↗](https://fungrim.org/entry/651a4a) --- $$\mathrm{Hypergeometric2F1Regularized}(a, b, -n, z)=\frac{\mathrm{Hypergeometric2F_1}(a+n+1, b+n+1, n+2, z)\mathrm{RisingFactorial}(a, n+1)\mathrm{RisingFactorial}(b, n+1)z^{n+1}}{(n+1)!}$$ **Holds when** $a\in\C\land b\in\C\land n\in\N\land z\in\C\setminus\lbrace1\rbrace$. **Symbols:** **Hypergeometric2F1Regularized** — Regularized Gauss hypergeometric function; **RisingFactorial** — Rising factorial. Used by the Compute Engine for simplification. [`65693e` · Fungrim entry ↗](https://fungrim.org/entry/65693e) --- $$\mathrm{Hypergeometric2F_1}(a, b, c, 1)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\setminus\Z_{\le0}\land\Re(c-a-b)\gt0$. Used by the Compute Engine for simplification. [`659ce8` · Fungrim entry ↗](https://fungrim.org/entry/659ce8) --- $$\frac{1}{\pi}(\sin(\pi(b-a))\mathrm{Hypergeometric2F1Regularized}(a, b, c, z))=\frac{\frac{\mathrm{Hypergeometric2F1Regularized}(a, a-c+1, a-b+1, 1/z)}{(-z)^{a}}}{\Gamma(b)\Gamma(c-a)}-\frac{\frac{\mathrm{Hypergeometric2F1Regularized}(b, b-c+1, -a+b+1, 1/z)}{(-z)^{b}}}{\Gamma(a)\Gamma(c-b)}$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\land z\in\C\land z\notin\lbrack0, \infty\rparen$. **Symbols:** **Hypergeometric2F1Regularized** — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. [`90ac58` · Fungrim entry ↗](https://fungrim.org/entry/90ac58) --- $$\mathrm{Hypergeometric2F_1}(1, 1, 2, z)=-(\frac{\ln(1-z)}{z})$$ **Holds when** $z\in\C\setminus\lbrace0, 1\rbrace$. Used by the Compute Engine for simplification. [`a85994` · Fungrim entry ↗](https://fungrim.org/entry/a85994) --- $$\mathrm{Hypergeometric2F1Regularized}(a, b, c, z)=\frac{\mathrm{Hypergeometric2F1Regularized}(a, c-b, c, \frac{z}{z-1})}{(1-z)^{a}}$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\land z\in\C\land z\notin\lbrack1, \infty\rparen$. **Symbols:** **Hypergeometric2F1Regularized** — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. [`b25089` · Fungrim entry ↗](https://fungrim.org/entry/b25089) --- $$\frac{1}{\pi}(\sin(\pi(-a-b+c))\mathrm{Hypergeometric2F1Regularized}(a, b, c, z))=\frac{\frac{\mathrm{Hypergeometric2F1Regularized}(a, a-c+1, a+b-c+1, 1-1/z)}{z^{a}}}{\Gamma(c-a)\Gamma(c-b)}-\frac{\mathrm{Hypergeometric2F1Regularized}(c-a, 1-a, -a-b+c+1, 1-1/z)z^{a-c}(1-z)^{-a-b+c}}{\Gamma(a)\Gamma(b)}$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\land z\in\C\land z\notin\lparen-\infty, 0\rbrack\land z\notin\lbrack1, \infty\rparen$. **Symbols:** **Hypergeometric2F1Regularized** — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. [`ca9123` · Fungrim entry ↗](https://fungrim.org/entry/ca9123) --- $$\frac{1}{\pi}(\sin(\pi(-a-b+c))\mathrm{Hypergeometric2F1Regularized}(a, b, c, z))=\frac{\mathrm{Hypergeometric2F1Regularized}(a, b, a+b-c+1, 1-z)}{\Gamma(c-a)\Gamma(c-b)}-\frac{\mathrm{Hypergeometric2F1Regularized}(c-a, c-b, -a-b+c+1, 1-z)(1-z)^{-a-b+c}}{\Gamma(a)\Gamma(b)}$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\land z\in\C\land z\notin\lparen-\infty, 0\rbrack\land z\notin\lbrack1, \infty\rparen$. **Symbols:** **Hypergeometric2F1Regularized** — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. [`db3eb9` · Fungrim entry ↗](https://fungrim.org/entry/db3eb9) --- $$(z(1-z)z\mapsto\mathrm{Hypergeometric2F_1}(a, b, c, z)^{\doubleprime}(z)+(c-(a+b+1)z)z\mapsto\mathrm{Hypergeometric2F_1}(a, b, c, z)^{\prime}(z))-ab\mathrm{Hypergeometric2F_1}(a, b, c, z)=0$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\setminus\Z_{\le0}\land z\in\C\setminus\lbrack1, \infty\rparen$. Used by the Compute Engine for simplification. [`f1bd89` · Fungrim entry ↗](https://fungrim.org/entry/f1bd89) --- $$\mathrm{Hypergeometric2F1Regularized}(a, b, c, z)=\frac{\mathrm{Hypergeometric2F_1}(a, b, c, z)}{\Gamma(c)}$$ **Holds when** $a\in\C\land b\in\C\land c\in\C\setminus\Z_{\le0}\land z\in\C$. **Symbols:** **Hypergeometric2F1Regularized** — Regularized Gauss hypergeometric function. Used by the Compute Engine for simplification. [`fe6e74` · Fungrim entry ↗](https://fungrim.org/entry/fe6e74) ---