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21: 8.15 Sums

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8.15.1 $$\gamma (a,\lambda x)={\lambda }^a\sum _{k=0}^{\mathrm{\infty }}\gamma (a+k,x)\frac{{(1-\lambda )}^k}{k!}.$$

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Symbols:

$!$: factorial (as in $n!$), $\gamma (a,z)$: incomplete gamma function, $x$: real variable, $a$: parameter and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.15.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.15 and Ch.8

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8.15.2 $$a\sum _{k=1}^{\mathrm{\infty }}\left(\frac{{\mathrm{e}}^{2\pi \mathrm{i}k(z+h)}}{{\left(2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,2\pi \mathrm{i}kz)+\frac{{\mathrm{e}}^{-2\pi \mathrm{i}k(z+h)}}{{\left(-2\pi \mathrm{i}k\right)}^{a+1}}\mathrm{\Gamma }(a,-2\pi \mathrm{i}kz)\right)=\zeta (-a,z+h)+\frac{z^{a+1}}{a+1}+\left(h-\frac{1}{2}\right)z^a,$$ $h\in [0,1]$.

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Symbols:

$\zeta (s,a)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

Proof sketch:

Start with and $z>0$. For $\mathrm{\Gamma }(a,\pm 2\pi \mathrm{i}kz)$ take integral representation (8.2.2) and use the substitution $t=\pm 2\pi \mathrm{i}kz(1+\tau )$. The sum and the integral can be interchanged, and the sum can be evaluated via (4.6.1). Use integration by parts. This will result in $\left(h-\frac{1}{2}\right)z^a$ plus two integrals with infinitely many poles. The residue theorem (§1.10(iv)) will give us an infinite series which can be identified via (25.11.1). For other values of $a$ and $z$ use analytic continuation.

Referenced by:

§25.11(x), §25.11(x), §8.15, Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/8.15.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

This equation was added.

See also:

Annotations for §8.15 and Ch.8

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22: 2.8 Differential Equations with a Parameter

§2.8 Differential Equations with a Parameter

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§2.8(i) Classification of Cases

… ►Zeros of $f(z)$ are also called turning points. … ►

§2.8(vi) Coalescing Transition Points

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23: 8.3 Graphics

… ►Some monotonicity properties of ${\gamma }^{\ast }(a,x)$ and $\mathrm{\Gamma }(a,x)$ in the four quadrants of the ($a,x$)-plane in Figure 8.3.6 are given in Erdélyi et al. (1953b, §9.6). … ►

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Figure 8.3.8: $\mathrm{\Gamma }(0.25,x+\mathrm{i}y)$, $-3\le x\le 3$, $-3\le y\le 3$. …There is a cut along the negative real axis. … Magnify 3D Help

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Figure 8.3.9: $\gamma (0.25,x+\mathrm{i}y)$, $-3\le x\le 3$, $-3\le y\le 3$. …There is a cut along the negative real axis. Magnify 3D Help

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Figure 8.3.14: $\mathrm{\Gamma }(2.5,x+\mathrm{i}y)$, $-2.2\le x\le 3$, $-3\le y\le 3$. …There is a cut along the negative real axis. … Magnify 3D Help

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Figure 8.3.15: $\gamma (2.5,x+\mathrm{i}y)$, $-2.2\le x\le 3$, $-3\le y\le 3$. …There is a cut along the negative real axis. Magnify 3D Help

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24: 8.11 Asymptotic Approximations and Expansions

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8.11.1 $$u_k={(-1)}^k{\left(1-a\right)}_k=(a-1)(a-2)\mathrm{\cdots }(a-k),$$

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Defines:

$u_k$ (locally)

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $a$: parameter and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/8.11.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(i), §8.11 and Ch.8

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8.11.2 $$\mathrm{\Gamma }(a,z)=z^{a-1}{\mathrm{e}}^{-z}\left(\sum _{k=0}^{n-1}\frac{u_k}{z^k}+R_n(a,z)\right),$$ $n=1,2,\mathrm{\dots }$.

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Symbols:

$\mathrm{e}$: base of natural logarithm, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter, $k$: nonnegative integer, $n$: nonnegative integer, $u_k$ and $R_n(a,z)$: remainder

A&S Ref:

6.5.32

Permalink:

http://dlmf.nist.gov/8.11.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(i), §8.11 and Ch.8

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8.11.3 $$R_n(a,z)=O\left(z^{-n}\right),$$ $|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter, $n$: nonnegative integer, $\delta$: small positive constant and $R_n(a,z)$: remainder

Permalink:

http://dlmf.nist.gov/8.11.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(i), §8.11 and Ch.8

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8.11.4 $$\gamma (a,z)=z^a{\mathrm{e}}^{-z}\sum _{k=0}^{\mathrm{\infty }}\frac{z^k}{{\left(a\right)}_{k+1}},$$ $a\ne 0,-1,-2,\mathrm{\dots }$.

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Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $\gamma (a,z)$: incomplete gamma function, $z$: complex variable, $a$: parameter and $k$: nonnegative integer

A&S Ref:

6.5.33 (corrected)

Referenced by:

§8.11(ii)

Permalink:

http://dlmf.nist.gov/8.11.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §8.11(ii), §8.11 and Ch.8

… ►If $z=\lambda a$, with $\lambda$ fixed, then as $a\to \mathrm{\infty }$ …

25: 28.1 Special Notation

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$m,n$	integers.
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$a,q,h$	real or complex parameters of Mathieu’s equation with $q=h^2$.
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… ►Alternative notations for the parameters $a$ and $q$ are shown in Table 28.1.1. …

26: 31.8 Solutions via Quadratures

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31.8.2 $$w_{\pm }(𝐦;\lambda ;z)=\sqrt{{\mathrm{\Psi }}_{g,N}(\lambda ,z)}\exp \left(\pm \frac{\mathrm{i}\nu (\lambda )}{2}{\int }_{z_0}^z\frac{t^{m_1}{(t-1)}^{m_2}{(t-a)}^{m_3}dt}{{\mathrm{\Psi }}_{g,N}(\lambda ,t)\sqrt{t(t-1)(t-a)}}\right)$$

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Symbols:

$dx$: differential of $x$ , $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $z$: complex variable, $\nu$: real or complex parameter, $m$: nonnegative integer, $a$: complex parameter, $\lambda =-4q$ , ${\mathrm{\Psi }}_{g,N}(\lambda ,z)$: polynomial, $g$: degree of $\lambda$ and $N$: degree of $z$

Referenced by:

§31.8

Permalink:

http://dlmf.nist.gov/31.8.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §31.8 and Ch.31

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27: 31.2 Differential Equations

… ►The parameters play different roles: $a$ is the singularity parameter; $\alpha ,\beta ,\gamma ,\delta ,ϵ$ are exponent parameters; $q$ is the accessory parameter. … ► $w(z)=z^{1-\gamma }{w}_1(z)$ satisfies (31.2.1) if $w_1$ is a solution of (31.2.1) with transformed parameters $q_1=q+(a\delta +ϵ)(1-\gamma )$; ${\alpha }_1=\alpha +1-\gamma$, ${\beta }_1=\beta +1-\gamma$, ${\gamma }_1=2-\gamma$. Next, $w(z)={(z-1)}^{1-\delta }{w}_2(z)$ satisfies (31.2.1) if $w_2$ is a solution of (31.2.1) with transformed parameters $q_2=q+a\gamma (1-\delta )$; ${\alpha }_2=\alpha +1-\delta$, ${\beta }_2=\beta +1-\delta$, ${\delta }_2=2-\delta$. … ►For example, if $\tilde{z}=z/a$, then the parameters are $\tilde{a}=1/a$, $\tilde{q}=q/a$; $\tilde{\delta }=ϵ$, $\widetilde{ϵ}=\delta$. …For example, $w(z)={(1-z)}^{-\alpha }\tilde{w}(z/(z-1))$, which arises from $\tilde{z}=z/(z-1)$, satisfies (31.2.1) if $\tilde{w}(\tilde{z})$ is a solution of (31.2.1) with $z$ replaced by $\tilde{z}$ and transformed parameters $\tilde{a}=a/(a-1)$, $\tilde{q}=-(q-a\alpha \gamma )/(a-1)$; $\tilde{\beta }=\alpha +1-\delta$, $\tilde{\delta }=\alpha +1-\beta$. …

28: 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

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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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29: 28.14 Fourier Series

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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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30: Bille C. Carlson

… ►In his paper Lauricella’s hypergeometric function $F_D$ (1963), he defined the $R$-function, a multivariate hypergeometric function that is homogeneous in its variables, each variable being paired with a parameter. …