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11: 28.14 Fourier Series

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28.14.1 $${\mathrm{me}}_{\nu }(z,q)=\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_{2m}^{\nu }(q){\mathrm{e}}^{\mathrm{i}(\nu +2m)z},$$

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Symbols:

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

A&S Ref:

20.3.8 (in slightly different form)

Referenced by:

§28.22(ii)

Permalink:

http://dlmf.nist.gov/28.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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28.14.4 $$q{c}_{2m+2}-\left(a-{(\nu +2m)}^2\right)c_{2m}+q{c}_{2m-2}=0,$$ $a={\lambda }_{\nu }\left(q\right),c_{2m}=c_{2m}^{\nu }(q)$,

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Symbols:

${\lambda }_{\nu +2n}\left(q\right)$: eigenvalues of Mathieu equation, $m$: integer, $q=h^2$: parameter, $\nu$: complex parameter, $a$: parameter and $c_{2m}(q)$: coefficients

Referenced by:

item (d), item (d)

Permalink:

http://dlmf.nist.gov/28.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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28.14.5 $$\sum _{m=-\mathrm{\infty }}^{\mathrm{\infty }}{\left(c_{2m}^{\nu }(q)\right)}^2=1;$$

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Symbols:

$m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Referenced by:

item (d)

Permalink:

http://dlmf.nist.gov/28.14.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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28.14.7 $$c_{-2m}^{-\nu }(q)=c_{2m}^{\nu }(q),$$

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Symbols:

$m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.14.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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28.14.8 $$c_{2m}^{\nu }(-q)={(-1)}^m{c}_{2m}^{\nu }(q).$$

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Symbols:

$m$: integer, $q=h^2$: parameter, $\nu$: complex parameter and $c_{2m}(q)$: coefficients

Permalink:

http://dlmf.nist.gov/28.14.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.14 and Ch.28

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12: 15.7 Continued Fractions

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15.7.1 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=t_0-\frac{u_{1}z}{t_1-}\frac{u_{2}z}{t_2-}\frac{u_{3}z}{t_3-}\mathrm{\cdots },$$

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $t_n$: coefficient and $u_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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15.7.3 $$\frac{𝐅(a,b;c;z)}{𝐅(a,b+1;c+1;z)}=v_0-\frac{w_1}{v_1-}\frac{w_2}{v_2-}\frac{w_3}{v_3-}\mathrm{\cdots },$$

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter, $c$: real or complex parameter, $v_n$: coefficient and $w_n$: coefficient

Permalink:

http://dlmf.nist.gov/15.7.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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15.7.5 $$\frac{𝐅(a,b;c;z)}{𝐅(a+1,b+1;c+1;z)}=x_0+\frac{y_1}{x_1+}\frac{y_2}{x_2+}\frac{y_3}{x_3+}\mathrm{\cdots },$$

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Symbols:

$𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $y$: real variable, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Permalink:

http://dlmf.nist.gov/15.7.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §15.7 and Ch.15

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13: 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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14: 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 $$(0,1){\mathrm{𝐻𝑓}}_m(a,q_m;\alpha ,\beta ,\gamma ,\delta ;z),$$ $m=0,1,2,\mathrm{\dots }$.

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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, $m$: 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

►The eigenvalues $q_m$ satisfy the continued-fraction equation ►

31.4.2 $$q=\frac{a\gamma P_1}{Q_1+q-}\frac{R_1{P}_2}{Q_2+q-}\frac{R_2{P}_3}{Q_3+q-}\mathrm{\cdots },$$

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Symbols:

$\gamma$: real or complex parameter, $a$: complex parameter, $q$: real or complex parameter, $P_j$: coefficient, $Q_j$: coefficient and $R_j$: coefficient

Referenced by:

§31.18

Permalink:

http://dlmf.nist.gov/31.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.4 and Ch.31

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

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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, $m$: 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

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15: 20.16 Software

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§20.16(ii) Real Argument and Parameter

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§20.16(iii) Complex Argument and/or Parameter

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16: 31.2 Differential Equations

… ►All other homogeneous linear differential equations of the second order having four regular singularities in the extended complex plane, $ℂ\cup \{\mathrm{\infty }\}$, can be transformed into (31.2.1). ►The parameters play different roles: $a$ is the singularity parameter; $\alpha ,\beta ,\gamma ,\delta ,ϵ$ are exponent parameters; $q$ is the accessory parameter. The total number of free parameters is six. … ►Then (suppressing the parameter $k$) … ►Except for the identity automorphism, each alters the parameters.

17: 8.17 Incomplete Beta Functions

… ►Throughout §§8.17 and 8.18 we assume that $a>0$, $b>0$, and $0\le x\le 1$. … ►

8.17.4 $$I_x(a,b)=1-I_{1-x}(b,a).$$

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Symbols:

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

A&S Ref:

6.6.3 (Symmetry relation)

Referenced by:

§8.17(v), §8.18(i)

Permalink:

http://dlmf.nist.gov/8.17.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(i), §8.17 and Ch.8

… ►With $a>0$, $b>0$, and , … ►

8.17.13 $$(a+b)I_x(a,b)=a{I}_x(a+1,b)+b{I}_x(a,b+1),$$

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Symbols:

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

A&S Ref:

6.6.7

Permalink:

http://dlmf.nist.gov/8.17.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(iv), §8.17 and Ch.8

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8.17.16 $$a{I}_x(a+1,b)=(a+cx)I_x(a,b)-cx{I}_x(a-1,b),$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/8.17.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §8.17(iv), §8.17 and Ch.8

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18: 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. …

19: 28.27 Addition Theorems

… ►Addition theorems provide important connections between Mathieu functions with different parameters and in different coordinate systems. …

20: 31.12 Confluent Forms of Heun’s Equation

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31.12.1 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\gamma }{z}+\frac{\delta }{z-1}+ϵ\right)\frac{dw}{dz}+\frac{\alpha z-q}{z(z-1)}w=0.$$

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Symbols:

$\frac{df}{dx}$: derivative of $f$ with respect to $x$ , $z$: complex variable, $\gamma$: real or complex parameter, $\delta$: real or complex parameter, $ϵ$: real or complex parameter, $q$: real or complex parameter and $\alpha$: real or complex parameter

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.2 $$\frac{d^{2}w}{{dz}^2}+\left(\frac{\delta }{z^2}+\frac{\gamma }{z}+1\right)\frac{dw}{dz}+\frac{\alpha z-q}{z^2}w=0.$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/31.12.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.3 $$\frac{d^{2}w}{{dz}^2}-\left(\frac{\gamma }{z}+\delta +z\right)\frac{dw}{dz}+\frac{\alpha z-q}{z}w=0.$$

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Symbols:

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

Referenced by:

Erratum (V1.0.7) for Equation (31.12.3)

Permalink:

http://dlmf.nist.gov/31.12.E3

Encodings:

pMML, png, TeX

Errata (effective with 1.0.7):

Originally the sign in front of the second term in this equation was $+$. The correct sign is $-$.

Reported 2013-10-31 by Henryk Witek

See also:

Annotations for §31.12, §31.12 and Ch.31

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31.12.4 $$\frac{d^{2}w}{{dz}^2}+\left(\gamma +z\right)z\frac{dw}{dz}+\left(\alpha z-q\right)w=0.$$

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Symbols:

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

Referenced by:

§31.12

Permalink:

http://dlmf.nist.gov/31.12.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §31.12, §31.12 and Ch.31

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