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11: 8.11 Asymptotic Approximations and Expansions

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8.11.10 $$P(a+1,x)=\frac{1}{2}\mathrm{erfc}\left(-y\right)-\frac{1}{3}\sqrt{\frac{2}{\pi a}}(1+y^2){\mathrm{e}}^{-y^2}+O\left(a^{-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(iv), §8.11 and Ch.8

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8.11.11 $${\gamma }^{\ast }(1-a,-x)=x^{a-1}\left(-\cos \left(\pi a\right)+\frac{\sin \left(\pi a\right)}{\pi }\left(2\sqrt{\pi }F\left(y\right)+\frac{2}{3}\sqrt{\frac{2\pi }{a}}\left(1-y^2\right)\right){\mathrm{e}}^{y^2}+O\left(a^{-1}\right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, ${\gamma }^{\ast }(a,z)$: incomplete gamma function, $\sin z$: sine function, $x$: real variable, $a$: parameter and $y$: change of variable

Permalink:

http://dlmf.nist.gov/8.11.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(iv), §8.11 and Ch.8

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12: 29.2 Differential Equations

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29.2.4 $$(1-k^2{\cos }^2\varphi )\frac{d^{2}w}{{d\varphi }^2}+k^2\cos \varphi \sin \varphi \frac{dw}{d\varphi }+(h-\nu (\nu +1)k^2{\cos }^2\varphi )w=0,$$

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Symbols:

$\cos z$: cosine function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sin z$: sine function, $h$: real parameter, $k$: real parameter, $\nu$: real parameter and $\varphi$: change of variable

Permalink:

http://dlmf.nist.gov/29.2.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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29.2.5 $$\varphi =\frac{1}{2}\pi -\mathrm{am}(z,k).$$

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Defines:

$\varphi$: change of variable (locally)

Symbols:

$\mathrm{am}(x,k)$: Jacobi’s amplitude function, $\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $k$: real parameter

Referenced by:

§29.15(i), §29.15(ii), §29.6(i)

Permalink:

http://dlmf.nist.gov/29.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §29.2(ii), §29.2 and Ch.29

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13: 19.25 Relations to Other Functions

… ►The three changes of parameter of $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$ in §19.7(iii) are unified in (19.21.12) by way of (19.25.14). …

14: 19.8 Quadratic Transformations

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19.8.14 $$2(k^2-{\alpha }^2)\mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\frac{{\omega }^2-{\alpha }^2}{1+k^{\prime }}\mathrm{\Pi }({\varphi }_1,{\alpha }_1^2,k_1)+k^{2}F(\varphi ,k)-(1+k^{\prime }){\alpha }_1^2{R}_C(c_1,c_1-{\alpha }_1^2),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter, $k_1$: change of variable, ${\varphi }_1$: change of variable, $\omega$: change of variable, ${\alpha }_1$: change of variable and $c_1$: change of variable

Permalink:

http://dlmf.nist.gov/19.8.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(ii), §19.8(ii), §19.8 and Ch.19

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19.8.20 $$\rho \mathrm{\Pi }(\varphi ,{\alpha }^2,k)=\frac{4}{1+k^{\prime }}\mathrm{\Pi }({\psi }_1,{\alpha }_1^2,k_1)+(\rho -1)F(\varphi ,k)-R_C(c-1,c-{\alpha }^2),$$

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Symbols:

$R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $F(\varphi ,k)$: Legendre’s incomplete elliptic integral of the first kind, $\mathrm{\Pi }(\varphi ,{\alpha }^2,k)$: Legendre’s incomplete elliptic integral of the third kind, $\varphi$: real or complex argument, $k$: real or complex modulus, $k^{\prime }$: complementary modulus, ${\alpha }^2$: real or complex parameter, $k_1$: change of variable, ${\psi }_1$: change of variable, $\rho$: change of variable, ${\alpha }_1^2$: change of variable and $c$: change of variable

Referenced by:

§19.36(ii)

Permalink:

http://dlmf.nist.gov/19.8.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §19.8(iii), §19.8 and Ch.19

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15: 15.8 Transformations of Variable

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15.8.32 $$\frac{{\left(1-z^3\right)}^a}{{\left(-z\right)}^{3a}}\left(\frac{1}{\mathrm{\Gamma }\left(a+\frac{2}{3}\right)\mathrm{\Gamma }\left(\frac{2}{3}\right)}F(\genfrac{}{}{0pt}{}{a,a+\frac{1}{3}}{\frac{2}{3}};z^{-3})+\frac{{\mathrm{e}}^{\frac{1}{3}\pi \mathrm{i}}}{z\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(\frac{4}{3}\right)}F(\genfrac{}{}{0pt}{}{a+\frac{1}{3},a+\frac{2}{3}}{\frac{4}{3}};z^{-3})\right)=\frac{3^{\frac{3}{2}a+\frac{1}{2}}{\mathrm{e}}^{\frac{1}{2}a\pi \mathrm{i}}\mathrm{\Gamma }\left(a+\frac{1}{3}\right){(1-\zeta )}^a}{2\pi \mathrm{\Gamma }\left(2a+\frac{2}{3}\right){(-\zeta )}^{2a}}F(\genfrac{}{}{0pt}{}{a+\frac{1}{3},3a}{2a+\frac{2}{3}};{\zeta }^{-1}),$$ $|z|>1$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $F(a,b;c;z)$ or $F(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}F_1^{}(a,b;c;z)$ Gauss’ hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $z$: complex variable, $a$: real or complex parameter and $\zeta$: change of variable

Referenced by:

§15.4(iii)

Permalink:

http://dlmf.nist.gov/15.8.E32

Encodings:

pMML, png, TeX

See also:

Annotations for §15.8(v), §15.8(v), §15.8 and Ch.15

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16: 18.9 Recurrence Relations and Derivatives

… ►The following three formulas change the degree but preserve the parameters, see (18.2.42)–(18.2.44) for similar formulas for more general OP’s. …

17: 28.2 Definitions and Basic Properties

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28.2.2 $$\zeta (1-\zeta )w^{\prime \prime }+\frac{1}{2}(1-2\zeta )w^{\prime }+\frac{1}{4}(a-2q(1-2\zeta ))w=0.$$

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Symbols:

$q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $\zeta$: change of variable

Permalink:

http://dlmf.nist.gov/28.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

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28.2.3 $$(1-{\zeta }^2)w^{\prime \prime }-\zeta w^{\prime }+\left(a+2q-4q{\zeta }^2\right)w=0.$$

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Symbols:

$q=h^2$: parameter, $a$: parameter, $w(z)$: Mathieu’s equation solution and $\zeta$: change of variable

A&S Ref:

20.1.7 (in slightly different notation)

Permalink:

http://dlmf.nist.gov/28.2.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.2(i), §28.2 and Ch.28

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18: 20.10 Integrals

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§20.10(i) Mellin Transforms with respect to the Lattice Parameter

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§20.10(ii) Laplace Transforms with respect to the Lattice Parameter

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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19: 3.8 Nonlinear Equations

… ►Then the sensitivity of a simple zero $z$ to changes in $\alpha$ is given by …

20: 31.7 Relations to Other Functions

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31.7.1 $${}_2{}^{}F_1^{}(\alpha ,\beta ;\gamma ;z)=H\mathrm{\ell }(1,\alpha \beta ;\alpha ,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(0,0;\alpha ,\beta ,\gamma ,\alpha +\beta +1-\gamma ;z)=H\mathrm{\ell }(a,a\alpha \beta ;\alpha ,\beta ,\gamma ,\alpha +\beta +1-\gamma ;z).$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $a$: complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.7(i), §31.7 and Ch.31

►Other reductions of $H\mathrm{\ell }$ to a ${}_2{}^{}F_1^{}$, with at least one free parameter, exist iff the pair $(a,p)$ takes one of a finite number of values, where $q=\alpha \beta p$. Below are three such reductions with three and two parameters. … ►

31.7.2 $$H\mathrm{\ell }(2,\alpha \beta ;\alpha ,\beta ,\gamma ,\alpha +\beta -2\gamma +1;z)={}_2{}^{}F_1^{}(\frac{1}{2}\alpha ,\frac{1}{2}\beta ;\gamma ;1-{(1-z)}^2),$$

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Symbols:

${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.7.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.7(i), §31.7 and Ch.31

… ►With $z={\mathrm{sn}}^2(\zeta ,k)$ and …