conjugate%20Poisson%20integral

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21: 14.30 Spherical and Spheroidal Harmonics

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14.30.6

Y_{{l},{-m}}\left(\theta,\phi\right)=(-1)^{m}\overline{Y_{{l},{m}}\left(\theta% ,\phi\right)}.

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Symbols:: $\overline{\NVar{z}}$ : complex conjugate, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $m$ : integer and $l$ : nonnegative integer
Referenced by:: §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.30.E6
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.19):: As a notational clarification, the spherical harmonic on the right-hand side has been explicity marked up as a complex conjugate.
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

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14.30.8

\int_{0}^{2\pi}\!\!\int_{0}^{\pi}\overline{Y_{{l_{1}},{m_{1}}}\left(\theta,% \phi\right)}Y_{{l_{2}},{m_{2}}}\left(\theta,\phi\right)\sin\theta\,\mathrm{d}% \theta\,\mathrm{d}\phi=\delta_{l_{1},l_{2}}\delta_{m_{1},m_{2}}.

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Symbols:: $\delta_{\NVar{j},\NVar{k}}$ : Kronecker delta, $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $m$ : integer and $l$ : nonnegative integer
Referenced by:: §14.30(ii)
Permalink:: http://dlmf.nist.gov/14.30.E8
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.19):: As a notational clarification, the first spherical harmonic in the integral over $\theta$ been explicity marked up as a complex conjugate.
See also:: Annotations for §14.30(ii), §14.30(ii), §14.30 and Ch.14

►See also (34.3.22), and for further related integrals see Askey et al. (1986). … ►

14.30.9

\mathsf{P}_{l}\left(\cos\theta_{1}\cos\theta_{2}+\sin\theta_{1}\sin\theta_{2}% \cos\left(\phi_{1}-\phi_{2}\right)\right)=\frac{4\pi}{2l+1}\sum_{m=-l}^{l}% \overline{Y_{{l},{m}}\left(\theta_{1},\phi_{1}\right)}Y_{{l},{m}}\left(\theta_% {2},\phi_{2}\right).

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\overline{\NVar{z}}$ : complex conjugate, $\cos\NVar{z}$ : cosine function, $\mathsf{P}_{\NVar{\nu}}\left(\NVar{x}\right)=\mathsf{P}^{0}_{\nu}\left(x\right)$ : Ferrers function of the first kind, $\sin\NVar{z}$ : sine function, $Y_{{\NVar{l}},{\NVar{m}}}\left(\NVar{\theta},\NVar{\phi}\right)$ : spherical harmonic, $m$ : integer and $l$ : nonnegative integer
Referenced by:: §18.18(ii)
Permalink:: http://dlmf.nist.gov/14.30.E9
Encodings:: pMML, png, TeX
Notational Change (effective with 1.0.19):: As a notational clarification, the first spherical harmonic in the sum over $m$ has been explicity marked up as a complex conjugate.
See also:: Annotations for §14.30(iii), §14.30(iii), §14.30 and Ch.14

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22: 6.19 Tables

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§6.19(ii) Real Variables

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Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\operatorname{Si}\left(x\right)$ , $-x^{-2}\operatorname{Cin}\left(x\right)$ , $x^{-1}\operatorname{Ein}\left(x\right)$ , $-x^{-1}\operatorname{Ein}\left(-x\right)$ , $x=0(.01)0.5$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=0.5(.01)2$ ; $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x=2(.1)10$ ; $x\mathrm{f}\left(x\right)$ , $x^{2}\mathrm{g}\left(x\right)$ , $xe^{-x}\operatorname{Ei}\left(x\right)$ , $xe^{x}E_{1}\left(x\right)$ , $x^{-1}=0(.005)0.1$ ; $\operatorname{Si}\left(\pi x\right)$ , $\operatorname{Cin}\left(\pi x\right)$ , $x=0(.1)10$ . Accuracy varies but is within the range 8S–11S.

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Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$ , $\operatorname{Ci}\left(x\right)$ , $x=0(.5)20(2)30$ , 8D; $\operatorname{Ei}\left(x\right)$ , $E_{1}\left(x\right)$ , $x=[0,100]$ , 8S.

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•

Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $ze^{z}E_{1}\left(z\right)$ , $x=-19(1)20$ , $y=0(1)20$ , 6D; $e^{z}E_{1}\left(z\right)$ , $x=-4(.5)-2$ , $y=0(.2)1$ , 6D; $E_{1}\left(z\right)+\ln z$ , $x=-2(.5)2.5$ , $y=0(.2)1$ , 6D.

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Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$ , $\pm x=0.5,1,3,5,10,15,20,50,100$ , $y=0(.5)1(1)5(5)30,50,100$ , 8S.

23: 8 Incomplete Gamma and Related
Functions

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24: 28 Mathieu Functions and Hill’s Equation

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25: Software Index

… ► ►►►►►►►

	Open Source				With Book			Commercial
…
11.16(iii) Integrals of …					✓		✓		✓	✓	a
…
12.21(ii) $U\left(a,x\right)$ , $V\left(a,x\right)$ , $\overline{U}\left(a,x\right)$ , $W\left(a,x\right)$ , $x,a\in\mathbb{R}$		✓	a	✓		✓	✓		✓	✓	a
12.21(iii) $U\left(a,z\right)$ , $V\left(a,z\right)$ , $\overline{U}\left(a,z\right)$ , $W\left(a,z\right)$ , $z,a\in\mathbb{C}$			a	✓					✓	✓	a
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19 Elliptic Integrals
…
20 Theta Functions
…

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26: 1.9 Calculus of a Complex Variable

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Complex Conjugate

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1.9.11

\overline{z}=x-\mathrm{i}y,

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Defines:: $\overline{\NVar{z}}$ : complex conjugate
Symbols:: $\mathrm{i}$ : imaginary unit and $z$ : variable
A&S Ref:: 3.7.7
Permalink:: http://dlmf.nist.gov/1.9.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

… ►If

f(z(t_{0}))=\infty

a\leq t_{0}\leq b

, then the integral is defined analogously to the infinite integrals in §1.4(v). … ►

Cauchy’s Integral Formula

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Poisson Integral

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27: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►A complex linear vector space

V

is called an inner product space if an inner product

\left\langle u,v\right\rangle\in\mathbb{C}

is defined for all

u,v\in V

with the properties: (i)

\left\langle u,v\right\rangle

is complex linear in

u

; (ii)

\left\langle u,v\right\rangle=\overline{\left\langle v,u\right\rangle}

; (iii)

\left\langle v,v\right\rangle\geq 0

; (iv) if

\left\langle v,v\right\rangle=0

then

v=0

. … ►where the integral kernel is given by … ►Should an eigenvalue correspond to more than a single linearly independent eigenfunction, namely a multiplicity greater than one, all such eigenfunctions will always be implied as being part of any sums or integrals over the spectrum. … ►Also, because

q

is real-valued,

f\in N_{z}

iff

\overline{f}\in N_{\overline{z}}

. … ►Integral transforms (10.22.78) and (10.22.79) are examples of the utility of these extensions. …

28: 23 Weierstrass Elliptic and Modular
Functions

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29: 12.11 Zeros

… ►When

a>-\frac{1}{2}

U\left(a,z\right)

has a string of complex zeros that approaches the ray

\operatorname{ph}z=\frac{3}{4}\pi

z\to\infty

, and a conjugate string. When

a>-\frac{1}{2}

the zeros are asymptotically given by

z_{a,s}

and

\overline{z_{a,s}}

, where

s

is a large positive integer and … ►

12.11.9

u_{a,1}\sim 2^{\frac{1}{2}}\mu\left(1-1.85575\;708\mu^{-4/3}-0.34438\;34\mu^{-% 8/3}-0.16871\;5\mu^{-4}-0.11414\mu^{-16/3}-0.0808\mu^{-20/3}-\cdots\right),

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Symbols:: $\sim$ : Poincaré asymptotic expansion, $a$ : real or complex parameter and $u_{a,s}$ : zeros
Referenced by:: §12.11(iii)
Permalink:: http://dlmf.nist.gov/12.11.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §12.11(iii), §12.11 and Ch.12

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30: 3.8 Nonlinear Equations

… ►Let

z^{2}-sz-t

be an approximation to the real quadratic factor of

p(z)

that corresponds to a pair of conjugate complex zeros or to a pair of real zeros. … ►

3.8.15

p(x)=(x-1)(x-2)\cdots(x-20)

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Permalink:: http://dlmf.nist.gov/3.8.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

… ►Consider

x=20

and

j=19

. We have

p^{\mspace{1.0mu}\prime}(20)=19!

and

a_{19}=1+2+\dots+20=210

. … ►

3.8.16

\frac{\mathrm{d}x}{\mathrm{d}a_{19}}=-\frac{20^{19}}{19!}=(-4.30\dots)\times 1% 0^{7}.

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Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ and $!$ : factorial (as in $n!$ )
Permalink:: http://dlmf.nist.gov/3.8.E16
Encodings:: pMML, png, TeX
See also:: Annotations for §3.8(vi), §3.8(vi), §3.8 and Ch.3

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