complex variable and parameters

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21: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.10 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\alpha a,\alpha b}{\alpha c};𝐓)$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable and $\alpha \to \mathrm{\infty }$: parameter

Permalink:

http://dlmf.nist.gov/35.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(iv), §35.7 and Ch.35

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35.7.11 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐈-{\alpha }^{-1}𝐓)$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $c$: complex variable and $\alpha \to \mathrm{\infty }$: parameter

Permalink:

http://dlmf.nist.gov/35.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(iv), §35.7 and Ch.35

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22: 10.34 Analytic Continuation

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10.34.1 $$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{m\nu \pi \mathrm{i}}{I}_{\nu }\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.30

Referenced by:

§10.30(ii), §10.34, §10.47(v)

Permalink:

http://dlmf.nist.gov/10.34.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.2 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)={\mathrm{e}}^{-m\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)-\pi \mathrm{i}\sin \left(m\nu \pi \right)\csc \left(\nu \pi \right)I_{\nu }\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.31

Referenced by:

§10.27, §10.34, §10.34

Permalink:

http://dlmf.nist.gov/10.34.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.3 $$I_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=(\mathrm{i}/\pi )\left(\pm {\mathrm{e}}^{m\nu \pi \mathrm{i}}{K}_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp {\mathrm{e}}^{(m\mp 1)\nu \pi \mathrm{i}}{K}_{\nu }\left(z\right)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i), §10.40(iii), §10.41(iv)

Permalink:

http://dlmf.nist.gov/10.34.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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10.34.4 $$K_{\nu }\left(z{\mathrm{e}}^{m\pi \mathrm{i}}\right)=\csc \left(\nu \pi \right)\left(\pm \sin \left(m\nu \pi \right)K_{\nu }\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)\mp \sin \left((m\mp 1)\nu \pi \right)K_{\nu }\left(z\right)\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\sin z$: sine function, $m$: integer, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.34, §10.40(i)

Permalink:

http://dlmf.nist.gov/10.34.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.34 and Ch.10

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23: 10.29 Recurrence Relations and Derivatives

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10.29.5 $${𝒵}_{\nu }^{(k)}\left(z\right)=\frac{1}{2^k}\left({𝒵}_{\nu -k}\left(z\right)+\left(\genfrac{}{}{0.0pt}{}{k}{1}\right){𝒵}_{\nu -k+2}\left(z\right)+\left(\genfrac{}{}{0.0pt}{}{k}{2}\right){𝒵}_{\nu -k+4}\left(z\right)+\mathrm{\cdots }+{𝒵}_{\nu +k}\left(z\right)\right).$$

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Symbols:

$\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, ${𝒵}_{\nu }\left(z\right)$: modified cylinder function, $k$: nonnegative integer, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.29

Referenced by:

§10.29(ii)

Permalink:

http://dlmf.nist.gov/10.29.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.29(ii), §10.29 and Ch.10

24: 10.39 Relations to Other Functions

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10.39.5 $$I_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }{\mathrm{e}}^{\pm z}}{\mathrm{\Gamma }\left(\nu +1\right)}M(\nu +\frac{1}{2},2\nu +1,\mp 2z),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M(a,b,z)$: $={}_1{}^{}F_1^{}(a;b;z)$ Kummer confluent hypergeometric function, $\mathrm{e}$: base of natural logarithm, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.39

Permalink:

http://dlmf.nist.gov/10.39.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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10.39.6 $$K_{\nu }\left(z\right)={\pi }^{\frac{1}{2}}{(2z)}^{\nu }{\mathrm{e}}^{-z}U(\nu +\frac{1}{2},2\nu +1,2z),$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.40(iv), (9.6.21), (9.6.22)

Permalink:

http://dlmf.nist.gov/10.39.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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10.39.7 $$I_{\nu }\left(z\right)=\frac{{(2z)}^{-\frac{1}{2}}{M}_{0,\nu }\left(2z\right)}{2^{2\nu }\mathrm{\Gamma }\left(\nu +1\right)},$$ $2\nu \ne -1,-2,-3,\mathrm{\dots }$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.47

Referenced by:

(9.6.23), (9.6.24)

Permalink:

http://dlmf.nist.gov/10.39.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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10.39.8 $$K_{\nu }\left(z\right)={\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{W}_{0,\nu }\left(2z\right).$$

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Symbols:

$W_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.48

Referenced by:

(9.6.21), (9.6.22)

Permalink:

http://dlmf.nist.gov/10.39.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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10.39.10 $$I_{\nu }\left(z\right)={(\frac{1}{2}z)}^{\nu }lim𝐅(\lambda ,\mu ;\nu +1;z^2/(4\lambda \mu )),$$

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Symbols:

$I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable and $\nu$: complex parameter

Referenced by:

§10.39

Permalink:

http://dlmf.nist.gov/10.39.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §10.39, §10.39 and Ch.10

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25: 12.15 Generalized Parabolic Cylinder Functions

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12.15.1 $$\frac{d^{2}w}{{dz}^2}+\left(\nu +{\lambda }^{-1}-{\lambda }^{-2}{z}^{\lambda }\right)w=0$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable and $\nu$: real or complex parameter

Referenced by:

§12.15

Permalink:

http://dlmf.nist.gov/12.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §12.15 and Ch.12

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26: 31.1 Special Notation

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$x$, $y$	real variables.
$z$, $\zeta$, $w$, $W$	complex variables.
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$a$	complex parameter, $|a|\ge 1,a\ne 1$.
$q,\alpha ,\beta ,\gamma ,\delta ,ϵ,\nu$	complex parameters.

… ►Sometimes the parameters are suppressed.

27: 18.33 Polynomials Orthogonal on the Unit Circle

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18.33.22 $$p^{\ast }(z)\equiv z^n\overline{p({\overline{z}}^{-1})}=\sum _{k=0}^n\overline{c_{n-k}}{z}^k.$$

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Symbols:

$\overline{z}$: complex conjugate, $\equiv$: equals by definition, $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

Source:

Simon (2005a, (1.1.7))

Permalink:

http://dlmf.nist.gov/18.33.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

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28: 13.2 Definitions and Basic Properties

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13.2.11 $$U(a,-n,z)=z^{n+1}U(a+n+1,n+2,z).$$

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Symbols:

$U(a,b,z)$: Kummer confluent hypergeometric function, $n$: nonnegative integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/13.2.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.2(i), §13.2(i), §13.2 and Ch.13

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29: 10.28 Wronskians and Cross-Products

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10.28.1 $$𝒲\left\{I_{\nu }\left(z\right),I_{-\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)I_{-\nu -1}\left(z\right)-I_{\nu +1}\left(z\right)I_{-\nu }\left(z\right)=-2\sin \left(\nu \pi \right)/(\pi z),$$

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Symbols:

$𝒲$: Wronskian, $\pi$: the ratio of the circumference of a circle to its diameter, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\sin z$: sine function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.14

Permalink:

http://dlmf.nist.gov/10.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

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10.28.2 $$𝒲\left\{K_{\nu }\left(z\right),I_{\nu }\left(z\right)\right\}=I_{\nu }\left(z\right)K_{\nu +1}\left(z\right)+I_{\nu +1}\left(z\right)K_{\nu }\left(z\right)=1/z.$$

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Symbols:

$𝒲$: Wronskian, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.15

Permalink:

http://dlmf.nist.gov/10.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.28 and Ch.10

30: 8.2 Definitions and Basic Properties

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8.2.1 $$\gamma (a,z)={\int }_0^z{t}^{a-1}{\mathrm{e}}^{-t}dt,$$ $\mathrm{\Re }a>0$,

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Defines:

$\gamma (a,z)$: incomplete gamma function

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part, $z$: complex variable and $a$: parameter

Referenced by:

§8.2(iii), §8.6(i), §8.6(ii)

Permalink:

http://dlmf.nist.gov/8.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

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8.2.2 $$\mathrm{\Gamma }(a,z)={\int }_z^{\mathrm{\infty }}{t}^{a-1}{\mathrm{e}}^{-t}dt,$$

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Defines:

$\mathrm{\Gamma }(a,z)$: incomplete gamma function

Symbols:

$dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $a$: parameter

A&S Ref:

6.5.3

Referenced by:

(8.15.2), §8.2(i), §8.2(iii), §8.21(iii), §8.6(i)

Permalink:

http://dlmf.nist.gov/8.2.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

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8.2.5 $$P(a,z)+Q(a,z)=1.$$

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Symbols:

$P(a,z)$: normalized incomplete gamma function, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable and $a$: parameter

Permalink:

http://dlmf.nist.gov/8.2.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(i), §8.2 and Ch.8

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8.2.12 $$\frac{d^{2}w}{{dz}^2}+\left(1+\frac{1-a}{z}\right)\frac{dw}{dz}=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: parameter and $w$: solution

Permalink:

http://dlmf.nist.gov/8.2.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(iii), §8.2 and Ch.8

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8.2.13 $$\frac{d^{2}w}{{dz}^2}-\left(1+\frac{1-a}{z}\right)\frac{dw}{dz}+\frac{1-a}{z^2}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $a$: parameter and $w$: solution

Permalink:

http://dlmf.nist.gov/8.2.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §8.2(iii), §8.2 and Ch.8

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