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11: 28.10 Integral Equations

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28.10.2 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cosh \left(2h\sin z\sin t\right){\mathrm{ce}}_{2n}(t,h^2)dt=\frac{A_0^{2n}(h^2)}{{\mathrm{ce}}_{2n}(0,h^2)}{\mathrm{ce}}_{2n}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.21 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

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28.10.3 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sin \left(2h\cos z\cos t\right){\mathrm{ce}}_{2n+1}(t,h^2)dt=-\frac{h{A}_1^{2n+1}(h^2)}{{\mathrm{ce}}_{2n+1}^{\prime }(\frac{1}{2}\pi ,h^2)}{\mathrm{ce}}_{2n+1}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.20 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

►

28.10.4 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cos z\cos t\cosh \left(2h\sin z\sin t\right){\mathrm{ce}}_{2n+1}(t,h^2)dt=\frac{A_1^{2n+1}(h^2)}{2{\mathrm{ce}}_{2n+1}(0,h^2)}{\mathrm{ce}}_{2n+1}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $A_m(q)$: Fourier coefficient

A&S Ref:

20.7.21 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

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28.10.5 $$\frac{2}{\pi }{\int }_0^{\pi /2}\sinh \left(2h\sin z\sin t\right){\mathrm{se}}_{2n+1}(t,h^2)dt=\frac{h{B}_1^{2n+1}(h^2)}{{\mathrm{se}}_{2n+1}^{\prime }(0,h^2)}{\mathrm{se}}_{2n+1}(z,h^2),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.23 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

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28.10.8 $$\frac{2}{\pi }{\int }_0^{\pi /2}\cos z\cos t\sinh \left(2h\sin z\sin t\right){\mathrm{se}}_{2n+2}(t,h^2)dt=\frac{h{B}_2^{2n+2}(h^2)}{2{\mathrm{se}}_{2n+2}^{\prime }(0,h^2)}{\mathrm{se}}_{2n+2}(z,h^2).$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\sinh z$: hyperbolic sine function, $\int$: integral, $\sin z$: sine function, $h$: parameter, $n$: integer, $z$: complex variable and $B_m(q)$: Fourier coefficient

A&S Ref:

20.7.23 (in different form)

Permalink:

http://dlmf.nist.gov/28.10.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.10(i), §28.10 and Ch.28

…

12: 28.23 Expansions in Series of Bessel Functions

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28.23.2 $${\mathrm{me}}_{\nu }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n{c}_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

►

28.23.3 $${\mathrm{me}}_{\nu }^{\prime }(0,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\mathrm{i}\tanh z\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{(-1)}^n(\nu +2n)c_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, $\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.4 $${\mathrm{me}}_{\nu }(\frac{1}{2}\pi ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)={\mathrm{e}}^{\mathrm{i}\nu \pi /2}\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\sinh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

►

28.23.5 $${\mathrm{me}}_{\nu }^{\prime }(\frac{1}{2}\pi ,h^2){\mathrm{M}}_{\nu }^{(j)}(z,h)=\mathrm{i}{\mathrm{e}}^{\mathrm{i}\nu \pi /2}\coth z\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}(\nu +2n)c_{2n}^{\nu }(h^2){𝒞}_{\nu +2n}^{(j)}(2h\sinh z),$$

ⓘ

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\coth z$: hyperbolic cotangent function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer, $z$: complex variable, $\nu$: complex parameter, $c_{2m}(q)$: coefficients and ${𝒞}_{\mu }^{(j)}$: cylinder functions

Permalink:

http://dlmf.nist.gov/28.23.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

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28.23.6 $${\mathrm{Mc}}_{2m}^{(j)}(z,h)={(-1)}^m{\left({\mathrm{ce}}_{2m}(0,h^2)\right)}^{-1}\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}{(-1)}^{\mathrm{\ell }}{A}_{2\mathrm{\ell }}^{2m}(h^2){𝒞}_{2\mathrm{\ell }}^{(j)}(2h\cosh z),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\cosh z$: hyperbolic cosine function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $m$: integer, $h$: parameter, $j$: integer, $z$: complex variable, ${𝒞}_{\mu }^{(j)}$: cylinder functions and $A_m(q)$: Fourier coefficient

A&S Ref:

20.6.12

Referenced by:

§28.23

Permalink:

http://dlmf.nist.gov/28.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §28.23 and Ch.28

…

13: 8.18 Asymptotic Expansions of $I_x(a,b)$

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Symmetric Case

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8.18.9 $$I_x(a,b)\sim \frac{1}{2}\mathrm{erfc}\left(-\eta \sqrt{b/2}\right)+\frac{1}{\sqrt{2\pi (a+b)}}{\left(\frac{x}{x_0}\right)}^a{\left(\frac{1-x}{1-x_0}\right)}^b\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{c}_k(\eta )}{{(a+b)}^k},$$

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $k$: nonnegative integer, $a$: parameter, $b$: parameter, $x$: variable and $c_k(\eta )$: coefficients

Permalink:

http://dlmf.nist.gov/8.18.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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General Case

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Inverse Function

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8.18.18 $$I_x(a,b)=p,$$ $0\le p\le 1$,

ⓘ

Symbols:

$I_x(a,b)$: incomplete beta function, $p$: parameter, $a$: parameter, $b$: parameter and $x$: variable

Permalink:

http://dlmf.nist.gov/8.18.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §8.18(ii), §8.18(ii), §8.18 and Ch.8

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14: 28.20 Definitions and Basic Properties

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28.20.9 $${\mathrm{M}}_{\nu }^{(3)}(z,h)=H_{\nu }^{(1)}\left(2h\cosh z\right)\left(1+O\left(\mathrm{sech}z\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.20.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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28.20.10 $${\mathrm{M}}_{\nu }^{(4)}(z,h)=H_{\nu }^{(2)}\left(2h\cosh z\right)\left(1+O\left(\mathrm{sech}z\right)\right),$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel function), $\cosh z$: hyperbolic cosine function, $\mathrm{sech}z$: hyperbolic secant function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/28.20.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iii), §28.20 and Ch.28

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28.20.15 $${\mathrm{Mc}}_n^{(j)}(z,h)={\mathrm{M}}_n^{(j)}(z,h),$$ $n=0,1,\mathrm{\dots }$,

ⓘ

Defines:

${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function

Symbols:

${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $n$: integer, $j$: integer and $z$: complex variable

Referenced by:

§28.20(vii)

Permalink:

http://dlmf.nist.gov/28.20.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(iv), §28.20 and Ch.28

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28.20.17 $${\mathrm{Ie}}_n(z,h)={\mathrm{i}}^{-n}{\mathrm{Mc}}_n^{(1)}(z,\mathrm{i}h),$$

ⓘ

Defines:

${\mathrm{Ie}}_n(z,h)$: modified Mathieu function

Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.8.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(v), §28.20 and Ch.28

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28.20.18 $${\mathrm{Io}}_n(z,h)={\mathrm{i}}^{-n}{\mathrm{Ms}}_n^{(1)}(z,\mathrm{i}h),$$

ⓘ

Defines:

${\mathrm{Io}}_n(z,h)$: modified Mathieu function

Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $z$: complex variable

A&S Ref:

20.8.10 (in slightly different form)

Permalink:

http://dlmf.nist.gov/28.20.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §28.20(v), §28.20 and Ch.28

…

15: 31.4 Solutions Analytic at Two Singularities: Heun Functions

… ►For an infinite set of discrete values $q_m$, $m=0,1,2,\mathrm{\dots }$, of the accessory parameter $q$, the function $H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$ is analytic at $z=1$, and hence also throughout the disk . … ►

31.4.1 $$(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$.

ⓘ

Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.6

Permalink:

http://dlmf.nist.gov/31.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.4 and Ch.31

… ►

31.4.3 $$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$,

ⓘ

Symbols:

$(s_1,s_2){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter and $\beta$: real or complex parameter

Referenced by:

§31.4, §31.6

Permalink:

http://dlmf.nist.gov/31.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §31.4 and Ch.31

…

16: 36.3 Visualizations of Canonical Integrals

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…

Figure 36.3.2: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,3\right)|$. …

►

…

Figure 36.3.3: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,0\right)|$. …

►

…

Figure 36.3.4: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,-3\right)|$. …

►

…

Figure 36.3.5: Modulus of swallowtail canonical integral function $|{\mathrm{\Psi }}_3\left(x,y,-7.5\right)|$. …

►

…

Figure 36.3.6: Modulus of elliptic umbilic canonical integral function $|{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,0\right)|$. …

…

17: 28.28 Integrals, Integral Representations, and Integral Equations

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28.28.2 $$\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{ce}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{ce}}_n(\alpha ,h^2){\mathrm{Mc}}_n^{(1)}(z,h),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

A&S Ref:

20.7.34 20.7.35 20.7.36 20.7.37

Permalink:

http://dlmf.nist.gov/28.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

►

28.28.3 $$\frac{1}{2\pi }{\int }_0^{2\pi }{\mathrm{e}}^{2\mathrm{i}hw}{\mathrm{se}}_n(t,h^2)dt={\mathrm{i}}^n{\mathrm{se}}_n(\alpha ,h^2){\mathrm{Ms}}_n^{(1)}(z,h),$$

ⓘ

Symbols:

${\mathrm{se}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Ms}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\alpha$: parameter

Permalink:

http://dlmf.nist.gov/28.28.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

… ►

28.28.11 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{2\mathrm{i}h\cosh z\cosh t}{\mathrm{Ce}}_{\nu }(t,h^2)dt=\frac{1}{2}\pi \mathrm{i}{\mathrm{e}}^{\mathrm{i}\nu \pi }{\mathrm{ce}}_{\nu }(0,h^2){\mathrm{M}}_{\nu }^{(3)}(z,h),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{M}}_{\nu }^{(j)}(z,h)$: modified Mathieu function, $h$: parameter, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§28.28(i)

Permalink:

http://dlmf.nist.gov/28.28.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

… ►

28.28.15 $${\int }_0^{\mathrm{\infty }}\cos \left(2h\cos y\cosh t\right){\mathrm{Ce}}_{2n}(t,h^2)dt={(-1)}^{n+1}\frac{1}{2}\pi {\mathrm{Mc}}_{2n}^{(2)}(0,h){\mathrm{ce}}_{2n}(y,h^2),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, ${\mathrm{Mc}}_n^{(j)}(z,h)$: radial Mathieu function, $h$: parameter, $n$: integer and $y$: real variable

Permalink:

http://dlmf.nist.gov/28.28.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

►

28.28.16 $${\int }_0^{\mathrm{\infty }}\sin \left(2h\cos y\cosh t\right){\mathrm{Ce}}_{2n}(t,h^2)dt=-\frac{\pi A_0^{2n}(h^2)}{2{\mathrm{ce}}_{2n}(\frac{1}{2}\pi ,h^2)}\left({\mathrm{ce}}_{2n}(y,h^2)\mp \frac{2}{\pi C_{2n}(h^2)}{\mathrm{fe}}_{2n}(y,h^2)\right),$$

ⓘ

Symbols:

${\mathrm{ce}}_n(z,q)$: Mathieu function, ${\mathrm{fe}}_n(z,q)$: second solution, Mathieu’s equation, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\cosh z$: hyperbolic cosine function, $\int$: integral, ${\mathrm{Ce}}_{\nu }(z,q)$: modified Mathieu function, $\sin z$: sine function, $h$: parameter, $n$: integer, $y$: real variable, $A_m(q)$: Fourier coefficient and $C_n(q)$: factor

Permalink:

http://dlmf.nist.gov/28.28.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §28.28(i), §28.28 and Ch.28

…

18: 32.6 Hamiltonian Structure

… ►

32.6.21 $${(z{\sigma }^{\prime \prime }-{\sigma }^{\prime })}^2+2\left({({\sigma }^{\prime })}^2-{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2{z}^2\right)(z{\sigma }^{\prime }-2\sigma )+8{\kappa }_0{\kappa }_{\mathrm{\infty }}{\theta }_0{\theta }_{\mathrm{\infty }}z{\sigma }^{\prime }=4{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2({\theta }_0^2+{\theta }_{\mathrm{\infty }}^2)z^2.$$

ⓘ

Symbols:

$z$: real, $\sigma$: function, $\theta$: parameter and ${\kappa }_j$: parameters

Referenced by:

§32.6(iv)

Permalink:

http://dlmf.nist.gov/32.6.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

… ►

32.6.22 $$q=\frac{{\kappa }_0\left(z{\sigma }^{\prime \prime }-(2{\theta }_0+1){\sigma }^{\prime }+2{\kappa }_0{\kappa }_{\mathrm{\infty }}{\theta }_{\mathrm{\infty }}z\right)}{{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2{z}^2-{({\sigma }^{\prime })}^2},$$

ⓘ

Symbols:

$z$: real, $q$: coordinate, $\sigma$: function, $\theta$: parameter and ${\kappa }_j$: parameters

Permalink:

http://dlmf.nist.gov/32.6.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

… ►

32.6.30 $$q=\frac{{\eta }_0\left(\zeta {\sigma }^{\prime \prime }-2{\theta }_0{\sigma }^{\prime }+{\eta }_0{\eta }_{\mathrm{\infty }}{\theta }_{\mathrm{\infty }}\right)}{{\eta }_0^2{\eta }_{\mathrm{\infty }}^2-4{({\sigma }^{\prime })}^2},$$

ⓘ

Symbols:

$q$: coordinate, $\sigma$: function and $\theta$: parameter

Permalink:

http://dlmf.nist.gov/32.6.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

… ►

32.6.37 $${(z{\sigma }^{\prime \prime }-{\sigma }^{\prime })}^2+2{({\sigma }^{\prime })}^2(z{\sigma }^{\prime }-2\sigma )-4{\kappa }_0{\kappa }_{\mathrm{\infty }}(\theta +1){\theta }_{\mathrm{\infty }}z{\sigma }^{\prime }=4{\kappa }_0^2{\kappa }_{\mathrm{\infty }}^2{z}^2.$$

ⓘ

Symbols:

$z$: real, $\sigma$: function, $\theta$: parameter and ${\kappa }_j$: parameters

Referenced by:

§32.6(iv)

Permalink:

http://dlmf.nist.gov/32.6.E37

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

… ►

32.6.38 $$q={\kappa }_0\left(z{\sigma }^{\prime \prime }-\theta {\sigma }^{\prime }+2{\kappa }_0{\kappa }_{\mathrm{\infty }}z\right)/{({\sigma }^{\prime })}^2,$$

ⓘ

Symbols:

$z$: real, $q$: coordinate, $\sigma$: function, $\theta$: parameter and ${\kappa }_j$: parameters

Permalink:

http://dlmf.nist.gov/32.6.E38

Encodings:

pMML, png, TeX

See also:

Annotations for §32.6(iv), §32.6 and Ch.32

…

19: 16.2 Definition and Analytic Properties

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16.2.1 $${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a_1\right)}_k\mathrm{\cdots }{\left(a_p\right)}_k}{{\left(b_1\right)}_k\mathrm{\cdots }{\left(b_q\right)}_k}\frac{z^k}{k!}.$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§10.16, §10.16, §10.39, §16.2(iv), §16.2(ii), §16.2(iii), §16.2(iii), §16.2(iii), §16.5, §18.17(v), §18.17(vi), §18.17(vii), §18.20(ii), §18.26(i), §18.30(i), §18.38(ii)

Permalink:

http://dlmf.nist.gov/16.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §16.2(i), §16.2 and Ch.16

… ►However, when one or more of the top parameters $a_j$ is a nonpositive integer the series terminates and the generalized hypergeometric function is a polynomial in $z$. … ►

§16.2(v) Behavior with Respect to Parameters

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16.2.5 $${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)={}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)/\left(\mathrm{\Gamma }\left(b_1\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_q\right)\right)=\sum _{k=0}^{\mathrm{\infty }}\frac{{\left(a_1\right)}_k\mathrm{\cdots }{\left(a_p\right)}_k}{\mathrm{\Gamma }\left(b_1+k\right)\mathrm{\cdots }\mathrm{\Gamma }\left(b_q+k\right)}\frac{z^k}{k!};$$

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Defines:

${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}𝐅_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$: scaled (or Olver’s) generalized hypergeometric function

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};z)$: alternatively ${}_p{}^{}F_q^{}(𝐚;𝐛;z)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{𝐚}{𝐛};z)$
generalized hypergeometric function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $p$: nonnegative integer, $q$: nonnegative integer, $z$: complex variable, $a,a_1,\mathrm{\dots },a_p$: real or complex parameters and $b,b_1,\mathrm{\dots },b_q$: real or complex parameters

Referenced by:

§16.2(v)

Permalink:

http://dlmf.nist.gov/16.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §16.2(v), §16.2 and Ch.16

… ►When $p\le q+1$ and $z$ is fixed and not a branch point, any branch of ${}_p{}^{}𝐅_q^{}(𝐚;𝐛;z)$ is an entire function of each of the parameters $a_1,\mathrm{\dots },a_p,b_1,\mathrm{\dots },b_q$.

20: 30.6 Functions of Complex Argument

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30.6.3 $$𝒲\left\{{\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2),{\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)\right\}=\frac{{(-1)}^m(n+m)!}{(1-z^2)(n-m)!}{A}_n^m({\gamma }^2)A_n^{-m}({\gamma }^2),$$

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Symbols:

$𝒲$: Wronskian, $!$: factorial (as in $n!$), ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, $z$: complex variable, $m$: nonnegative integer, $n\ge m$: integer degree, $A_n^m({\gamma }^2)$ and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

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30.6.4 $${\mathrm{𝑃𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m{\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2),$$

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Symbols:

$\mathrm{i}$: imaginary unit, ${\mathrm{𝑃𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

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30.6.5 $${\mathrm{𝑄𝑠}}_n^m(x\pm \mathrm{i}0,{\gamma }^2)={(\mp \mathrm{i})}^m\left({\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)\mp \frac{1}{2}\mathrm{i}\pi {\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, ${\mathrm{𝑄𝑠}}_n^m(z,{\gamma }^2)$: spheroidal wave function of complex argument, ${\mathrm{𝖯𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the first kind, ${\mathrm{𝖰𝗌}}_n^m(x,{\gamma }^2)$: spheroidal wave function of the second kind, $x$: real variable, $m$: nonnegative integer, $n\ge m$: integer degree and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §30.6, §30.6 and Ch.30

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