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21: 2.6 Distributional Methods

… ►Furthermore,

K^{+}

contains the distributions

H

\delta

, and

t^{\lambda}

t>0

, for any real (or complex) number

\lambda

, where

H

is the distribution associated with the Heaviside function

H\left(t\right)

(§1.16(iv)), and

t^{\lambda}

is the distribution defined by (2.6.12)–(2.6.14), depending on the value of

\lambda

. …

22: 31.14 General Fuchsian Equation

… ►

31.14.1

{\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}+\left(\sum_{j=1}^{N}\frac{\gamma_% {j}}{z-a_{j}}\right)\frac{\mathrm{d}w}{\mathrm{d}z}+\left(\sum_{j=1}^{N}\frac{% q_{j}}{z-a_{j}}\right)w=0},

\sum_{j=1}^{N}q_{j}=0

ⓘ

Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities
Referenced by:: §31.15(i)
Permalink:: http://dlmf.nist.gov/31.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14 and Ch.31

… ►

31.14.3

w(z)=\left(\prod_{j=1}^{N}(z-a_{j})^{-\gamma_{j}/2}\right)W(z),

ⓘ

Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $W(z)$ : solution
Permalink:: http://dlmf.nist.gov/31.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

►

31.14.4

\frac{{\mathrm{d}}^{2}W}{{\mathrm{d}z}^{2}}=\sum_{j=1}^{N}\left(\frac{\tilde{% \gamma}_{j}}{(z-a_{j})^{2}}+\frac{\tilde{q}_{j}}{z-a_{j}}\right)W,

\sum_{j=1}^{N}\tilde{q}_{j}=0

ⓘ

Defines:: $W(z)$ : solution (locally)
Symbols:: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of $f$ with respect to $x$ , $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities
Permalink:: http://dlmf.nist.gov/31.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §31.14(i), §31.14(i), §31.14 and Ch.31

…

23: Bibliography

… ►

Arblib (C) Arb: A C Library for Arbitrary Precision Ball Arithmetic.

ⓘ

Notes:: An extended-precision C code for special functions. The library uses arbitrary-precision ball arithmetic. Ball arithmetic is an efficient technique for performing numerical computations with automatic and rigorous tracking of error bounds. Arb is designed with computer algebra and computational number theory in mind, extending the principles behind FLINT to the domain of real and complex numbers.
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

…

24: 1.10 Functions of a Complex Variable

… ►Let

\alpha

and

\beta

be real or complex numbers that are not integers. …

… ►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. …

26: 31.15 Stieltjes Polynomials

… ►

31.15.3

\sum_{j=1}^{N}\frac{\gamma_{j}}{t_{k}-a_{j}}+\sum_{j=1}^{n-1}\frac{1}{t_{k}-z_% {j}^{\prime}}=0.

ⓘ

Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $t_{k}$ : zero of $V(z)$
Permalink:: http://dlmf.nist.gov/31.15.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.6

a_{j}<a_{j+1},

j=1,2,\dots,N-1

ⓘ

Symbols:: $j$ : nonnegative integer, $a$ : complex parameter and $N+1$ : number of singularities
Referenced by:: §31.15(iii)
Permalink:: http://dlmf.nist.gov/31.15.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.7

q_{j}=\gamma_{j}\sum_{k=1}^{n}\frac{1}{z_{k}-a_{j}},

j=1,2,\dots,N

ⓘ

Symbols:: $z$ : complex variable, $\gamma$ : real or complex parameter, $j$ : nonnegative integer, $n$ : nonnegative integer, $a$ : complex parameter, $q$ : real or complex parameter and $N+1$ : number of singularities
Permalink:: http://dlmf.nist.gov/31.15.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(ii), §31.15 and Ch.31

… ►

31.15.8

S_{\mathbf{m}}(z_{1})S_{\mathbf{m}}(z_{2})\cdots S_{\mathbf{m}}(z_{N-1}),

z_{j}\in(a_{j},a_{j+1})

ⓘ

Symbols:: $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $z$ : complex variable, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $S_{\mathbf{m}}(z)$ : Stieltjes polynomial
Referenced by:: §31.15(iii)
Permalink:: http://dlmf.nist.gov/31.15.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(iii), §31.15 and Ch.31

►

31.15.9

S_{\mathbf{l}}(z_{1})S_{\mathbf{l}}(z_{2})\cdots S_{\mathbf{l}}(z_{N-1}),

z_{j}\in(a_{j},a_{j+1})

ⓘ

Symbols:: $\in$ : element of, $(\NVar{a},\NVar{b})$ : open interval, $z$ : complex variable, $j$ : nonnegative integer, $a$ : complex parameter, $N+1$ : number of singularities and $S_{\mathbf{m}}(z)$ : Stieltjes polynomial
Permalink:: http://dlmf.nist.gov/31.15.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §31.15(iii), §31.15 and Ch.31

…

27: Bibliography M

… ►

mpmath (free python library)

ⓘ

Notes:: Mpmath is a pure-Python library for multiprecision floating-point arithmetic. It provides an extensive set of transcendental functions, unlimited exponent sizes, complex numbers, interval arithmetic, numerical integration and differentiation, root-finding, linear algebra, and much more.
External Links:: mpmath
Referenced by:: § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index, § ‣ Software Cross Index.
Encodings:: BibTeX

…

28: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Conversely, if complex numbers

c_{n}

satisfy (1.18.5) then there is a unique

v\in V

such that (1.18.3) holds and

v

can be given by …

29: 9.9 Zeros

… ►However,

\operatorname{Bi}\left(z\right)

and

\operatorname{Bi}'\left(z\right)

each have an infinite number of complex zeros. …

30: 25.2 Definition and Expansions

… ►

25.2.9

\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}% {2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}% N^{1-s-2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{% \widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,

\Re s>-2n

;

n,N=1,2,3,\dots

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\Re$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Keywords:: Euler–Maclaurin formula, improper integral
Proof sketch:: Derivable from (25.2.8) by repeated integration by parts.
Referenced by:: §25.18(i), (25.2.10)
Permalink:: http://dlmf.nist.gov/25.2.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §25.2(iii), §25.2 and Ch.25

►

25.2.10

\zeta\left(s\right)=\frac{1}{s-1}+\frac{1}{2}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0% pt}{}{s+2k-2}{2k-1}\frac{B_{2k}}{2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{% 1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{x^{s+2n+1}}\,\mathrm{d}x,

\Re s>-2n

n=1,2,3,\dots

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\Re$ : real part, $k$ : nonnegative integer, $n$ : nonnegative integer, $x$ : real variable and $s$ : complex variable
Keywords:: Euler–Maclaurin formula
Proof sketch:: Derivable from (25.2.9) with $N=1$ .
A&S Ref:: 23.2.3
Permalink:: http://dlmf.nist.gov/25.2.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §25.2(iii), §25.2 and Ch.25

… ►

25.2.11

\zeta\left(s\right)=\prod_{p}(1-p^{-s})^{-1},

\Re s>1

ⓘ

Symbols:: $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\Re$ : real part, $p$ : prime number and $s$ : complex variable
Keywords:: infinite product, primes
Source:: Apostol (1976, p. 231)
A&S Ref:: 23.2.2
Referenced by:: §25.10(i)
Permalink:: http://dlmf.nist.gov/25.2.E11
Encodings:: pMML, png, TeX
See also:: Annotations for §25.2(iv), §25.2 and Ch.25

…