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21: 8.27 Approximations

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DiDonato (1978) gives a simple approximation for the function $F(p,x)=x^{-p}{\mathrm{e}}^{x^2/2}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-t^2/2}{t}^{p}dt$ (which is related to the incomplete gamma function by a change of variables) for real $p$ and large positive $x$. This takes the form $F(p,x)=4x/h(p,x)$, approximately, where $h(p,x)=3(x^2-p)+\sqrt{{(x^2-p)}^2+8(x^2+p)}$ and is shown to produce an absolute error $O\left(x^{-7}\right)$ as $x\to \mathrm{\infty }$.

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22: 31.2 Differential Equations

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31.2.8 $$\frac{d^{2}w}{{d\zeta }^2}+\left((2\gamma -1)\frac{\mathrm{cn}\zeta \mathrm{dn}\zeta }{\mathrm{sn}\zeta }-(2\delta -1)\frac{\mathrm{sn}\zeta \mathrm{dn}\zeta }{\mathrm{cn}\zeta }-(2ϵ-1)k^2\frac{\mathrm{sn}\zeta \mathrm{cn}\zeta }{\mathrm{dn}\zeta }\right)\frac{dw}{d\zeta }+4k^2(\alpha \beta {\mathrm{sn}}^2\zeta -q)w=0.$$

ⓘ

Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $\mathrm{dn}(z,k)$: Jacobian elliptic function, $\mathrm{sn}(z,k)$: Jacobian elliptic function, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter, $\zeta$: change of variable and $k$: modulus

Referenced by:

§31.7(ii)

Permalink:

http://dlmf.nist.gov/31.2.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §31.2(iv), §31.2(iv), §31.2 and Ch.31

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23: 2.4 Contour Integrals

… ►The change of integration variable is given by ►

2.4.18 $$p(\alpha ,t)=\frac{1}{3}{w}^3+a{w}^2+bw+c,$$

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Symbols:

$p(t)$: analytic function, $w$: change of variable, $a$ , $b$ and $c$: real constant

Permalink:

http://dlmf.nist.gov/2.4.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(v), §2.4 and Ch.2

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2.4.19 $$I(\alpha ,z)={\mathrm{e}}^{-cz}{\int }_{𝒬}\exp \left(-z\left(\frac{1}{3}{w}^3+a{w}^2+bw\right)\right)f(\alpha ,w)dw,$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $I(\alpha ,z)$: integral, $w$: change of variable, $a$ , $b$ , $𝒬$: countour, $f(\alpha ,w)$: function and $c$: real constant

Referenced by:

§2.4(v)

Permalink:

http://dlmf.nist.gov/2.4.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(v), §2.4 and Ch.2

… ►By making a further change of variable ►

2.4.21 $$w=z^{-1/3}v-a,$$

ⓘ

Symbols:

$w$: change of variable and $a$

Referenced by:

§2.4(v)

Permalink:

http://dlmf.nist.gov/2.4.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §2.4(v), §2.4 and Ch.2

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24: 29.6 Fourier Series

… ►With $\varphi =\frac{1}{2}\pi -\mathrm{am}(z,k)$, as in (29.2.5), we have ►

29.6.1 $${\mathrm{𝐸𝑐}}_{\nu }^{2m}(z,k^2)=\frac{1}{2}{A}_0+\sum _{p=1}^{\mathrm{\infty }}{A}_{2p}\cos \left(2p\varphi \right).$$

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Symbols:

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $\cos z$: cosine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $A_{2p}$: coefficients

Referenced by:

§29.15(i), §29.6(i)

Permalink:

http://dlmf.nist.gov/29.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(i), §29.6 and Ch.29

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29.6.16 $${\mathrm{𝐸𝑐}}_{\nu }^{2m+1}(z,k^2)=\sum _{p=0}^{\mathrm{\infty }}{A}_{2p+1}\cos \left((2p+1)\varphi \right).$$

ⓘ

Symbols:

${\mathrm{𝐸𝑐}}_{\nu }^m(z,k^2)$: Lamé function, $\cos z$: cosine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $A_{2p}$: coefficients

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.6.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(ii), §29.6 and Ch.29

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29.6.31 $${\mathrm{𝐸𝑠}}_{\nu }^{2m+1}(z,k^2)=\sum _{p=0}^{\mathrm{\infty }}{B}_{2p+1}\sin \left((2p+1)\varphi \right).$$

ⓘ

Symbols:

${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\sin z$: sine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $B_{2p+1}$: coefficents

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.6.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iii), §29.6 and Ch.29

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29.6.46 $${\mathrm{𝐸𝑠}}_{\nu }^{2m+2}(z,k^2)=\sum _{p=1}^{\mathrm{\infty }}{B}_{2p}\sin \left(2p\varphi \right).$$

ⓘ

Symbols:

${\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)$: Lamé function, $\sin z$: sine function, $m$: nonnegative integer, $p$: nonnegative integer, $z$: complex variable, $k$: real parameter, $\nu$: real parameter, $\varphi$: change of variable and $B_{2p+1}$: coefficents

Referenced by:

§29.15(i)

Permalink:

http://dlmf.nist.gov/29.6.E46

Encodings:

pMML, png, TeX

See also:

Annotations for §29.6(iv), §29.6 and Ch.29

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25: 2.3 Integrals of a Real Variable

… ►A uniform approximation can be constructed by quadratic change of integration variable: ►

2.3.25 $$p(\alpha ,t)=\frac{1}{2}{w}^2-aw+b,$$

ⓘ

Symbols:

$p(\alpha ,t)$: function, $w$: change of variable, $a(\alpha )$ and $b(\alpha )$

Permalink:

http://dlmf.nist.gov/2.3.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(v), §2.3 and Ch.2

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2.3.27 $$w={(2p(\alpha ,0)-2p(\alpha ,\alpha ))}^{1/2}\pm {(2p(\alpha ,t)-2p(\alpha ,\alpha ))}^{1/2},$$

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Symbols:

$p(\alpha ,t)$: function and $w$: change of variable

Permalink:

http://dlmf.nist.gov/2.3.E27

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(v), §2.3 and Ch.2

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2.3.30 $$f(\alpha ,w)=q(\alpha ,t){\left(\frac{t}{w}\right)}^{\lambda -1}\frac{dt}{dw},$$

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Defines:

$f(\alpha ,w)$: function (locally)

Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $q(\alpha ,t)$: function, $w$: change of variable and $\lambda$: positive constant

Permalink:

http://dlmf.nist.gov/2.3.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(v), §2.3 and Ch.2

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2.3.31 $$f(\alpha ,w)=\sum _{s=0}^{\mathrm{\infty }}{\varphi }_s(\alpha ){(w-a)}^s,$$

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Symbols:

$w$: change of variable, $a(\alpha )$ , $f(\alpha ,w)$: function and ${\varphi }_s(\alpha )$: coefficients

Permalink:

http://dlmf.nist.gov/2.3.E31

Encodings:

pMML, png, TeX

See also:

Annotations for §2.3(v), §2.3 and Ch.2

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26: 12.10 Uniform Asymptotic Expansions for Large Parameter

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12.10.35 $$U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim 2{\pi }^{\frac{1}{2}}{\mu }^{\frac{1}{3}}g(\mu )\varphi (\zeta )\left(\mathrm{Ai}\left({\mu }^{\frac{4}{3}}\zeta \right)\sum _{s=0}^{\mathrm{\infty }}\frac{A_s(\zeta )}{{\mu }^{4s}}+\frac{{\mathrm{Ai}}^{\prime }\left({\mu }^{\frac{4}{3}}\zeta \right)}{{\mu }^{\frac{8}{3}}}\sum _{s=0}^{\mathrm{\infty }}\frac{B_s(\zeta )}{{\mu }^{4s}}\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\zeta$: change of variable, $\varphi (\zeta )$: function, $A_s(\zeta )$: coefficients, $B_s(\zeta )$: coefficients and $g(\mu )$: expansion

Referenced by:

§12.10(vii), §12.10(viii)

Permalink:

http://dlmf.nist.gov/12.10.E35

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

►

12.10.36 $$U^{\prime }(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})\sim \frac{{(2\pi )}^{\frac{1}{2}}{\mu }^{\frac{2}{3}}g(\mu )}{\varphi (\zeta )}\left(\frac{\mathrm{Ai}\left({\mu }^{\frac{4}{3}}\zeta \right)}{{\mu }^{\frac{4}{3}}}\sum _{s=0}^{\mathrm{\infty }}\frac{C_s(\zeta )}{{\mu }^{4s}}+{\mathrm{Ai}}^{\prime }\left({\mu }^{\frac{4}{3}}\zeta \right)\sum _{s=0}^{\mathrm{\infty }}\frac{D_s(\zeta )}{{\mu }^{4s}}\right),$$

ⓘ

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $U(a,z)$: parabolic cylinder function, $s$: nonnegative integer, $\zeta$: change of variable, $\varphi (\zeta )$: function, $C_s(\zeta )$: coefficients, $D_s(\zeta )$: coefficients and $g(\mu )$: expansion

Referenced by:

§12.10(vii)

Permalink:

http://dlmf.nist.gov/12.10.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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12.10.40 $$\varphi (\zeta )={\left(\frac{\zeta }{t^2-1}\right)}^{\frac{1}{4}}.$$

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Defines:

$\varphi (\zeta )$: function (locally)

Symbols:

$\zeta$: change of variable

Referenced by:

§12.10(vii)

Permalink:

http://dlmf.nist.gov/12.10.E40

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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12.10.41 $$t=1+w-\frac{1}{10}{w}^2+\frac{11}{350}{w}^3-\frac{823}{63000}{w}^4+\frac{\mathrm{1\hspace{0.33em}50653}}{\mathrm{242\hspace{0.33em}55000}}{w}^5+\mathrm{\cdots },$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter and $\zeta$: change of variable

Referenced by:

§12.11(iii), §12.11(iii)

Permalink:

http://dlmf.nist.gov/12.10.E41

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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12.10.45 $$\chi (\zeta )=\frac{{\varphi }^{\prime }(\zeta )}{\varphi (\zeta )}=\frac{1-2t{(\varphi (\zeta ))}^6}{4\zeta }.$$

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Defines:

$\chi (\zeta )$ (locally)

Symbols:

$\zeta$: change of variable and $\varphi (\zeta )$: function

Permalink:

http://dlmf.nist.gov/12.10.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §12.10(vii), §12.10 and Ch.12

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27: 29.2 Differential Equations

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

ⓘ

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

ⓘ

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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28: 19.2 Definitions

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$k_c=k^{\prime },$

►

$p=1-{\alpha }^2,$

►

$x=\tan \varphi ,$

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29: 8.11 Asymptotic Approximations and Expansions

… ►If $x=a+{(2a)}^{\frac{1}{2}}y$ and $a\to +\mathrm{\infty }$, then ►

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

ⓘ

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

►

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

ⓘ

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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30: 27.14 Unrestricted Partitions

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27.14.8 $$p\left(n\right)\sim \frac{{\mathrm{e}}^{K\sqrt{n}}}{4n\sqrt{3}},$$

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Symbols:

$\sim$: asymptotic equality, $\mathrm{e}$: base of natural logarithm, $p\left(n\right)$: total number of partitions of $n$ , $n$: positive integer and $K$: change of variable

A&S Ref:

24.2.1 III

Permalink:

http://dlmf.nist.gov/27.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(iii), §27.14 and Ch.27

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27.14.9 $$p\left(n\right)=\frac{1}{\pi \sqrt{2}}\sum _{k=1}^{\mathrm{\infty }}\sqrt{k}{A}_k(n){\left[\frac{d}{dt}\frac{\sinh \left(K\sqrt{t}/k\right)}{\sqrt{t}}\right]}_{t=n-(1/24)},$$

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\sinh z$: hyperbolic sine function, $p\left(n\right)$: total number of partitions of $n$ , $k$: positive integer, $n$: positive integer, $K$: change of variable and $A_k(n)$: coefficients

A&S Ref:

24.2.1 I.C

Referenced by:

§27.20

Permalink:

http://dlmf.nist.gov/27.14.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(iii), §27.14 and Ch.27

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