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21: 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:

pMML, png, TeX

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:

pMML, png, TeX

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

Encodings:

pMML, png, TeX

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

Encodings:

pMML, png, TeX

See also:

Annotations for §28.19, §28.19 and Ch.28

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22: 31.5 Solutions Analytic at Three Singularities: Heun Polynomials

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31.5.1

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Symbols:

$\gamma$: real or complex parameter, $n$: nonnegative integer, $a$: complex parameter, $P_j$: coefficient, $Q_j$: coefficient and $R_j$: coefficient

Permalink:

http://dlmf.nist.gov/31.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.5 and Ch.31

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31.5.2 $${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)=H\mathrm{\ell }(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$$

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Defines:

${\mathrm{𝐻𝑝}}_{n,m}(a,q_{n,m};-n,\beta ,\gamma ,\delta ;z)$: Heun polynomials

Symbols:

$H\mathrm{\ell }(a,q;\alpha ,\beta ,\gamma ,\delta ;z)$: Heun functions, $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $m$: nonnegative integer, $n$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter and $\beta$: real or complex parameter

Permalink:

http://dlmf.nist.gov/31.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.5 and Ch.31

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23: 31.6 Path-Multiplicative Solutions

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31.6.1 $$(s_1,s_2){\mathrm{𝐻𝑓}}_m^{\nu }(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$,

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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, $\nu$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $q$: real or complex parameter, $\alpha$: real or complex parameter, $\beta$: real or complex parameter and $s_j$: parameters

Permalink:

http://dlmf.nist.gov/31.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.6 and Ch.31

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24: 9.14 Incomplete Airy Functions

… ►Incomplete Airy functions are defined by the contour integral (9.5.4) when one of the integration limits is replaced by a variable real or complex parameter. …

25: 32.1 Special Notation

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$m,n$	integers.
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$z$	complex variable.
…

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26: 4.14 Definitions and Periodicity

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4.14.4 $$\tan z=\frac{\sin z}{\cos z},$$

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Defines:

$\tan z$: tangent function

Symbols:

$\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.3

Referenced by:

§4.14, §4.45(i)

Permalink:

http://dlmf.nist.gov/4.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4.14.5 $$\csc z=\frac{1}{\sin z},$$

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Defines:

$\csc z$: cosecant function

Symbols:

$\sin z$: sine function and $z$: complex variable

A&S Ref:

4.3.4

Permalink:

http://dlmf.nist.gov/4.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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4.14.6 $$\sec z=\frac{1}{\cos z},$$

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Defines:

$\sec z$: secant function

Symbols:

$\cos z$: cosine function and $z$: complex variable

A&S Ref:

4.3.5

Permalink:

http://dlmf.nist.gov/4.14.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

… ►In $ℂ$ the zeros of $\sin z$ are $z=k\pi$, $k\in \mathbb{Z}$; the zeros of $\cos z$ are $z=\left(k+\frac{1}{2}\right)\pi$, $k\in \mathbb{Z}$. … ►

4.14.8 $$\sin \left(z+2k\pi \right)=\sin z,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\sin z$: sine function, $k$: integer and $z$: complex variable

A&S Ref:

4.3.7

Permalink:

http://dlmf.nist.gov/4.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.14 and Ch.4

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27: 7.25 Software

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§7.25(iii) $\mathrm{erf}z$, $\mathrm{erfc}z$, $w\left(z\right)$, $z\in ℂ$

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§7.25(v) $C\left(z\right)$, $S\left(z\right)$, $z\in ℂ$

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§7.25(vii) $ℱ\left(z\right)$, $G\left(z\right)$, $z\in ℂ$

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28: 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:

pMML, png, TeX

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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12.4.4 $$u_2(a,z)={\mathrm{e}}^{-\frac{1}{4}{z}^2}\left(z+(a+\frac{3}{2})\frac{z^3}{3!}+(a+\frac{3}{2})(a+\frac{7}{2})\frac{z^5}{5!}+\mathrm{\cdots }\right).$$

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Defines:

$u_2(a,z)$: 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:

19.2.3 (modification of)

Permalink:

http://dlmf.nist.gov/12.4.E4

Encodings:

pMML, png, TeX

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:

pMML, png, TeX

See also:

Annotations for §12.4 and Ch.12

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29: 4.7 Derivatives and Differential Equations

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4.7.8 $$\frac{d}{dz}{\mathrm{e}}^{az}=a{\mathrm{e}}^{az},$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\mathrm{e}$: base of natural logarithm, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.50 (gives n-th derivative.)

Permalink:

http://dlmf.nist.gov/4.7.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(ii), §4.7 and Ch.4

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4.7.9 $$\frac{d}{dz}{a}^z=a^z\ln a,$$ $a\ne 0$.

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $\ln z$: principal branch of logarithm function, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.51

Permalink:

http://dlmf.nist.gov/4.7.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(ii), §4.7 and Ch.4

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4.7.10 $$\frac{d}{dz}{z}^a=a{z}^{a-1},$$

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Symbols:

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

A&S Ref:

4.2.52

Permalink:

http://dlmf.nist.gov/4.7.E10

Encodings:

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See also:

Annotations for §4.7(ii), §4.7 and Ch.4

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4.7.14 $$\frac{d^{2}w}{{dz}^2}=aw,$$ $a\ne 0$,

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Symbols:

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

Permalink:

http://dlmf.nist.gov/4.7.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(ii), §4.7 and Ch.4

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4.7.15 $$w=A{\mathrm{e}}^{\sqrt{a}z}+B{\mathrm{e}}^{-\sqrt{a}z},$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $a$: real or complex constant, $z$: complex variable, $A$: arbitrary constant and $B$: arbitrary constant

Permalink:

http://dlmf.nist.gov/4.7.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §4.7(ii), §4.7 and Ch.4

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30: 10.13 Other Differential Equations

… ►In the following equations $\nu ,\lambda ,p,q$, and $r$ are real or complex constants with $\lambda \ne 0$, $p\ne 0$, and $q\ne 0$. ►

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

Encodings:

pMML, png, TeX

See also:

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:

pMML, png, TeX

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:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

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10.13.6 $$w^{\prime \prime }+({\lambda }^2{\mathrm{e}}^{2z}-{\nu }^2)w=0,$$ $w={𝒞}_{\nu }\left(\lambda {\mathrm{e}}^z\right)$,

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $\mathrm{e}$: base of natural logarithm, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.54

Referenced by:

§10.36

Permalink:

http://dlmf.nist.gov/10.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §10.13 and Ch.10

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