complex variables and parameters

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1: 32.1 Special Notation

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$m,n$	integers.
$x$	real variable.
$z$	complex variable.
$k$	real parameter.

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2: 25.1 Special Notation

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$k,m,n$	nonnegative integers.
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$x$	real variable.
$a$	real or complex parameter.
$s=\sigma +\mathrm{i}t$	complex variable.
$z=x+\mathrm{i}y$	complex variable.
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3: 28.19 Expansions in Series of ${\mathrm{me}}_{\nu +2n}$ Functions

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28.19.1 $$f(z+\pi )={\mathrm{e}}^{\mathrm{i}\nu \pi }f(z).$$

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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, $z$: complex variable, $\nu$: complex parameter and $f(z)$: function

Permalink:

http://dlmf.nist.gov/28.19.E1

Encodings:

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See also:

Annotations for §28.19 and Ch.28

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28.19.2 $$f(z)=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{f}_n{\mathrm{me}}_{\nu +2n}(z,q),$$

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Defines:

$f(z)$: function (locally)

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f_n$: coefficients

Referenced by:

§28.19

Permalink:

http://dlmf.nist.gov/28.19.E2

Encodings:

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See also:

Annotations for §28.19 and Ch.28

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28.19.3 $$f_n=\frac{1}{\pi }{\int }_0^{\pi }f(z){\mathrm{me}}_{\nu +2n}(-z,q)dz.$$

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Defines:

$f_n$: coefficients (locally)

Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $f(z)$: function

Permalink:

http://dlmf.nist.gov/28.19.E3

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See also:

Annotations for §28.19 and Ch.28

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28.19.4 $${\mathrm{e}}^{\mathrm{i}\nu z}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{-2n}^{\nu +2n}(q){\mathrm{me}}_{\nu +2n}(z,q),$$

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Symbols:

${\mathrm{me}}_n(z,q)$: Mathieu function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $q=h^2$: parameter, $n$: integer, $z$: complex variable, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.19.E4

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See also:

Annotations for §28.19, §28.19 and Ch.28

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4: 10.13 Other Differential Equations

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10.13.1 $$w^{\prime \prime }+\left({\lambda }^2-\frac{{\nu }^2-\frac{1}{4}}{z^2}\right)w=0,$$ $w=z^{\frac{1}{2}}{𝒞}_{\nu }\left(\lambda z\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.49

Referenced by:

§1.18(vi), §10.36

Permalink:

http://dlmf.nist.gov/10.13.E1

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Annotations for §10.13 and Ch.10

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10.13.2 $$w^{\prime \prime }+\left(\frac{{\lambda }^2}{4z}-\frac{{\nu }^2-1}{4z^2}\right)w=0,$$ $w=z^{\frac{1}{2}}{𝒞}_{\nu }\left(\lambda z^{\frac{1}{2}}\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.50

Permalink:

http://dlmf.nist.gov/10.13.E2

Encodings:

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See also:

Annotations for §10.13 and Ch.10

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10.13.5 $$z^2{w}^{\prime \prime }+(1-2r)z{w}^{\prime }+({\lambda }^2{q}^2{z}^{2q}+r^2-{\nu }^2{q}^2)w=0,$$ $w=z^r{𝒞}_{\nu }\left(\lambda z^q\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.53

Permalink:

http://dlmf.nist.gov/10.13.E5

Encodings:

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See also:

Annotations for §10.13 and Ch.10

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10.13.9 $$z^2{w}^{\prime \prime \prime }+3z{w}^{\prime \prime }+(4z^2+1-4{\nu }^2)w^{\prime }+4zw=0,$$ $w={𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.58 (modified)

Referenced by:

§10.13, §10.36

Permalink:

http://dlmf.nist.gov/10.13.E9

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Annotations for §10.13 and Ch.10

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10.13.10 $$z^3{w}^{\prime \prime \prime }+z(4z^2+1-4{\nu }^2)w^{\prime }+(4{\nu }^2-1)w=0,$$ $w=z{𝒞}_{\nu }\left(z\right){𝒟}_{\nu }(z)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $z$: complex variable, $\nu$: complex parameter and ${𝒟}_{\nu }(z)$: cylinder function

A&S Ref:

9.1.59

Permalink:

http://dlmf.nist.gov/10.13.E10

Encodings:

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See also:

Annotations for §10.13 and Ch.10

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5: 12.8 Recurrence Relations and Derivatives

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12.8.1 $$zU(a,z)-U(a-1,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

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Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.4

Referenced by:

§12.8(i)

Permalink:

http://dlmf.nist.gov/12.8.E1

Encodings:

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See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.2 $$U^{\prime }(a,z)+\frac{1}{2}zU(a,z)+(a+\frac{1}{2})U(a+1,z)=0,$$

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Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.1

Permalink:

http://dlmf.nist.gov/12.8.E2

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Annotations for §12.8(i), §12.8 and Ch.12

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12.8.3 $$U^{\prime }(a,z)-\frac{1}{2}zU(a,z)+U(a-1,z)=0,$$

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Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.2

Permalink:

http://dlmf.nist.gov/12.8.E3

Encodings:

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See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.4 $$2U^{\prime }(a,z)+U(a-1,z)+(a+\frac{1}{2})U(a+1,z)=0.$$

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Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.3

Referenced by:

§12.8(i)

Permalink:

http://dlmf.nist.gov/12.8.E4

Encodings:

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See also:

Annotations for §12.8(i), §12.8 and Ch.12

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12.8.5 $$zV(a,z)-V(a+1,z)+(a-\frac{1}{2})V(a-1,z)=0,$$

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Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.6.8

Permalink:

http://dlmf.nist.gov/12.8.E5

Encodings:

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See also:

Annotations for §12.8(i), §12.8 and Ch.12

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6: 33.13 Complex Variable and Parameters

§33.13 Complex Variable and Parameters

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7: 12.4 Power-Series Expansions

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12.4.1 $$U(a,z)=U(a,0)u_1(a,z)+U^{\prime }(a,0)u_2(a,z),$$

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Symbols:

$U(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $u_1(a,z)$: solution and $u_2(a,z)$: solution

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.4.E1

Encodings:

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See also:

Annotations for §12.4 and Ch.12

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12.4.2 $$V(a,z)=V(a,0)u_1(a,z)+V^{\prime }(a,0)u_2(a,z),$$

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Symbols:

$V(a,z)$: parabolic cylinder function, $z$: complex variable, $a$: real or complex parameter, $u_1(a,z)$: solution and $u_2(a,z)$: solution

Referenced by:

§12.20

Permalink:

http://dlmf.nist.gov/12.4.E2

Encodings:

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See also:

Annotations for §12.4 and Ch.12

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12.4.3 $$u_1(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(1+(a+\frac{1}{2})\frac{z^2}{2!}+(a+\frac{1}{2})(a+\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

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Defines:

$u_1(a,z)$: solution (locally)

Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable and $a$: real or complex parameter

A&S Ref:

19.2.1 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E3

Encodings:

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See also:

Annotations for §12.4 and Ch.12

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12.4.5 $$u_1(a,z)={\mathrm{e}}^{\frac{1}{4}{z}^2}\left(1+(a-\frac{1}{2})\frac{z^2}{2!}+(a-\frac{1}{2})(a-\frac{5}{2})\frac{z^4}{4!}+\mathrm{\cdots }\right),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $a$: real or complex parameter and $u_1(a,z)$: solution

A&S Ref:

19.2.2 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E5

Encodings:

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See also:

Annotations for §12.4 and Ch.12

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12.4.6 $$u_2(a,z)={\mathrm{e}}^{\frac{1}{4}{z}^2}\left(z+(a-\frac{3}{2})\frac{z^3}{3!}+(a-\frac{3}{2})(a-\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $z$: complex variable, $a$: real or complex parameter and $u_2(a,z)$: solution

A&S Ref:

19.2.4 (modification of)

Referenced by:

§12.14(v), §12.7(iv)

Permalink:

http://dlmf.nist.gov/12.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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8: 12.1 Special Notation

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$x,y$	real variables.
$z$	complex variable.
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$a,\nu$	real or complex parameters.
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►Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. …

9: 8.4 Special Values

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8.4.5 $$\mathrm{\Gamma }(1,z)={\mathrm{e}}^{-z},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/8.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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8.4.9 $$P(n+1,z)=1-{\mathrm{e}}^{-z}{e}_n(z),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $P(a,z)$: normalized incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions

A&S Ref:

6.5.13

Permalink:

http://dlmf.nist.gov/8.4.E9

Encodings:

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See also:

Annotations for §8.4 and Ch.8

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8.4.10 $$Q(n+1,z)={\mathrm{e}}^{-z}{e}_n(z),$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $Q(a,z)$: normalized incomplete gamma function, $z$: complex variable, $n$: nonnegative integer and $e_n(z)$: functions

Permalink:

http://dlmf.nist.gov/8.4.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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8.4.11 $$e_n(z)=\sum _{k=0}^n\frac{z^k}{k!}.$$

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Defines:

$e_n(z)$: functions (locally)

Symbols:

$!$: factorial (as in $n!$), $z$: complex variable, $k$: nonnegative integer and $n$: nonnegative integer

A&S Ref:

6.5.11

Referenced by:

§8.11(v), §8.7

Permalink:

http://dlmf.nist.gov/8.4.E11

Encodings:

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See also:

Annotations for §8.4 and Ch.8

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8.4.12 $${\gamma }^{\ast }(-n,z)=z^n,$$

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Symbols:

${\gamma }^{\ast }(a,z)$: incomplete gamma function, $z$: complex variable and $n$: nonnegative integer

A&S Ref:

6.5.14

Referenced by:

§8.13(i)

Permalink:

http://dlmf.nist.gov/8.4.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.4 and Ch.8

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10: 10.10 Continued Fractions

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10.10.1 $$\frac{J_{\nu }\left(z\right)}{J_{\nu -1}\left(z\right)}=\frac{1}{2\nu z^{-1}-}\frac{1}{2(\nu +1)z^{-1}-}\frac{1}{2(\nu +2)z^{-1}-}\mathrm{\cdots },$$ $z\ne 0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.73

Referenced by:

§10.33, §10.74(v)

Permalink:

http://dlmf.nist.gov/10.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §10.10 and Ch.10

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10.10.2 $$\frac{J_{\nu }\left(z\right)}{J_{\nu -1}\left(z\right)}=\frac{\frac{1}{2}z/\nu }{1-}\frac{\frac{1}{4}{z}^2/(\nu (\nu +1))}{1-}\frac{\frac{1}{4}{z}^2/((\nu +1)(\nu +2))}{1-}\mathrm{\cdots },$$ $\nu \ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.73

Referenced by:

§10.33, §10.74(v)

Permalink:

http://dlmf.nist.gov/10.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.10 and Ch.10

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