DLMF: Untitled Document

… ►

See accompanying text — Figure 22.3.26: Density plot of $|\operatorname{sn}\left(5,k\right)|$ as a function of complex $k^{2}$ , $-10\leq\Re\left(k^{2}\right)\leq 20$ , $-10\leq\Im\left(k^{2}\right)\leq 10$ . … Magnify

►

… ►

•

Zhang and Jin (1996, pp. 185–195) tabulates $J_{n}\left(x\right)$ , $J_{n}'\left(x\right)$ , $Y_{n}\left(x\right)$ , $Y_{n}'\left(x\right)$ , $n=0(1)10(10)50,100$ , $x=1$ , 5, 10, 25, 50, 100, 9S; $J_{n+\alpha}\left(x\right)$ , $J_{n+\alpha}'\left(x\right)$ , $Y_{n+\alpha}\left(x\right)$ , $Y_{n+\alpha}'\left(x\right)$ , $n=0(1)5,10,30,50,100$ , $\alpha=\tfrac{1}{4},\tfrac{1}{3},\tfrac{1}{2},\tfrac{2}{3},\tfrac{3}{4}$ , $x=1,5,10,50$ , 8S; real and imaginary parts of $J_{n+\alpha}\left(z\right)$ , $J_{n+\alpha}'\left(z\right)$ , $Y_{n+\alpha}\left(z\right)$ , $Y_{n+\alpha}'\left(z\right)$ , $n=0(1)15,20(10)50,100$ , $\alpha=0,\tfrac{1}{2}$ , $z=4+2i$ , $20+10i$ , 8S.

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•

Zhang and Jin (1996, pp. 240–250) tabulates $I_{n}\left(x\right)$ , $I_{n}'\left(x\right)$ , $K_{n}\left(x\right)$ , $K_{n}'\left(x\right)$ , $n=0(1)10(10)50,100$ , $x=1,5,10,25,50,100$ , 9S; $I_{n+\alpha}\left(x\right)$ , $I_{n+\alpha}'\left(x\right)$ , $K_{n+\alpha}\left(x\right)$ , $K_{n+\alpha}'\left(x\right)$ , $n=0(1)5$ , 10, 30, 50, 100, $\alpha=\tfrac{1}{4}$ , $\tfrac{1}{3}$ , $\tfrac{1}{2}$ , $\tfrac{2}{3}$ , $\tfrac{3}{4}$ , $x=1$ , 5, 10, 50, 8S; real and imaginary parts of $I_{n+\alpha}\left(z\right)$ , $I_{n+\alpha}'\left(z\right)$ , $K_{n+\alpha}\left(z\right)$ , $K_{n+\alpha}'\left(z\right)$ , $n=0(1)15$ , 20(10)50, 100, $\alpha=0,\tfrac{1}{2}$ , $z=4+2i,20+10i$ , 8S.

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•

Zhang and Jin (1996, pp. 296–305) tabulates $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{j}_{n}'\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ , $\mathsf{y}_{n}'\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $\mathsf{k}_{n}'\left(x\right)$ , $n=0(1)10(10)30$ , 50, 100, $x=1$ , 5, 10, 25, 50, 100, 8S; $x\mathsf{j}_{n}\left(x\right)$ , $(x\mathsf{j}_{n}\left(x\right))^{\prime}$ , $x\mathsf{y}_{n}\left(x\right)$ , $(x\mathsf{y}_{n}\left(x\right))^{\prime}$ (Riccati–Bessel functions and their derivatives), $n=0(1)10(10)30$ , 50, 100, $x=1$ , 5, 10, 25, 50, 100, 8S; real and imaginary parts of $\mathsf{j}_{n}\left(z\right)$ , $\mathsf{j}_{n}'\left(z\right)$ , $\mathsf{y}_{n}\left(z\right)$ , $\mathsf{y}_{n}'\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}\left(z\right)$ , ${\mathsf{i}^{(1)}_{n}}'\left(z\right)$ , $\mathsf{k}_{n}\left(z\right)$ , $\mathsf{k}_{n}'\left(z\right)$ , $n=0(1)15$ , 20(10)50, 100, $z=4+2i$ , $20+10i$ , 8S. (For the notation replace $j, y, i, k$ by $\mathsf{j}$ , $\mathsf{y}$ , ${\mathsf{i}^{(1)}}$ , $\mathsf{k}$ , respectively.)

…

… ►

20.10.1

\int_{0}^{\infty}x^{s-1}\theta_{2}\left(0\middle|ix^{2}\right)\,\mathrm{d}x=2^% {s}(1-2^{-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>1

,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E1
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>1$ because $\theta_{2}\left(0\middle|ix^{2}\right)=x^{-1}\theta_{4}\left(0\middle|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$ , which follows from (20.7.31) and (20.4.5).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.2

\int_{0}^{\infty}x^{s-1}(\theta_{3}\left(0\middle|ix^{2}\right)-1)\,\mathrm{d}% x=\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>1

,

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E2
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>1$ because $\theta_{3}\left(0\middle|ix^{2}\right)=x^{-1}\theta_{3}\left(0\middle|ix^{-2}% \right)\sim x^{-1}$ as $x\to 0+$ , which follows from (20.7.32) and (20.4.4).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.3

\int_{0}^{\infty}x^{s-1}(1-\theta_{4}\left(0\middle|ix^{2}\right))\,\mathrm{d}% x=(1-2^{1-s})\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta\left(s\right),

\Re s>0

.

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\Re$ : real part
Keywords:: Mellin transform
Referenced by:: §20.10(i), Erratum (V1.1.5) for §20.10(i)
Permalink:: http://dlmf.nist.gov/20.10.E3
Encodings:: pMML, png, TeX
Modification (effective with 1.1.5):: Originally the constraint was $\Re s>2$ . This can be relaxed to $\Re s>0$ because $1-\theta_{4}\left(0\middle|ix^{2}\right)=1-x^{-1}\theta_{2}\left(0\middle|ix^{% -2}\right)\sim 1$ as $x\to 0+$ , which follows from (20.7.33) and (20.4.3).
See also:: Annotations for §20.10(i), §20.10 and Ch.20

… ►Let

s

,

\ell

, and

\beta

be constants such that

\Re s>0

,

\ell>0

, and

\left|\Re\beta\right|+\left|\Im\beta\right|\leq\ell

. … ►

20.10.4

\int_{0}^{\infty}e^{-st}\theta_{1}\left(\frac{\beta\pi}{2\ell}\middle|\frac{i% \pi t}{\ell^{2}}\right)\,\mathrm{d}t=\int_{0}^{\infty}e^{-st}\theta_{2}\left(% \frac{(1+\beta)\pi}{2\ell}\middle|\frac{i\pi t}{\ell^{2}}\right)\,\mathrm{d}t=% -\frac{\ell}{\sqrt{s}}\sinh\left(\beta\sqrt{s}\right)\operatorname{sech}\left(% \ell\sqrt{s}\right),

ⓘ

Symbols:: $\theta_{\NVar{j}}\left(\NVar{z}\middle|\NVar{\tau}\right)$ : theta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\operatorname{sech}\NVar{z}$ : hyperbolic secant function, $\sinh\NVar{z}$ : hyperbolic sine function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $s$ : parameter, $\beta$ : parameter and $\ell$ : parameter
Permalink:: http://dlmf.nist.gov/20.10.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §20.10(ii), §20.10 and Ch.20

…

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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.

►

•

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.

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	Open Source								With Book				Commercial
…
10.77(vi) Bessel Functions–Imaginary Order and Real Argument	✓	✓					✓	✓						✓	✓	✓
…
16.27(ii) Real Arguments	✓		✓	✓	✓		a	✓						✓	✓	✓
…
20 Theta Functions
…
22.22(ii) Real Argument	✓		✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓	✓
…

… ►

Commercial Software.

Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

…

… ►

•

Zhang and Jin (1996, p. 337) tabulates $\operatorname{Ai}\left(x\right)$ , $\operatorname{Ai}'\left(x\right)$ , $\operatorname{Bi}\left(x\right)$ , $\operatorname{Bi}'\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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•

Woodward and Woodward (1946) tabulates the real and imaginary parts of $\operatorname{Ai}\left(z\right)$ , $\operatorname{Ai}'\left(z\right)$ , $\operatorname{Bi}\left(z\right)$ , $\operatorname{Bi}'\left(z\right)$ for $\Re z=-2.4(.2)2.4$ , $\Im z=-2.4(.2)0$ . Precision is 4D.

►

•

Harvard University (1945) tabulates the real and imaginary parts of $h_{1}(z)$ , $h_{1}^{\prime}(z)$ , $h_{2}(z)$ , $h_{2}^{\prime}(z)$ for $-x_{0}\leq\Re z\leq x_{0}$ , $0\leq\Im z\leq y_{0}$ , $|x_{0}+iy_{0}|<6.1$ , with interval 0.1 in $\Re z$ and $\Im z$ . Precision is 8D. Here $h_{1}(z)=-2^{4/3}3^{1/6}i\operatorname{Ai}\left(e^{-\pi i/3}z\right)$ , $h_{2}(z)=2^{4/3}3^{1/6}i\operatorname{Ai}\left(e^{\pi i/3}z\right)$ .

… ►

•

Corless et al. (1992) gives the real and imaginary parts of $\beta_{k}$ for $k=1(1)13$ ; 14S.

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•

Nosova and Tumarkin (1965) tabulates $e_{0}(x)\equiv\pi\operatorname{Hi}\left(-x\right)$ , $e^{\prime}_{0}(x)=-\pi\operatorname{Hi}'\left(-x\right)$ , $\widetilde{e}_{0}(-x)\equiv-\pi\operatorname{Gi}\left(x\right)$ , $\widetilde{e}^{\mspace{2.0mu}\prime}_{0}(-x)=\pi\operatorname{Gi}'\left(x\right)$ for $x=-1(.01)10$ ; 7D. Also included are the real and imaginary parts of $e_{0}(z)$ and $ie^{\prime}_{0}(z)$ , where $z=iy$ and $y=0(.01)9$ ; 6-7D.

…

… ►When

z=e^{i\theta}

,

0\leq\theta\leq 2\pi

, (25.12.1) becomes … ►The series also converges when

|z|=1

, provided that

\Re s>1

. … ►valid when

\Re s>0

and

\left|\operatorname{ph}\left(1-z\right)\right|<\pi

, or

\Re s>1

and

z=1

. … ►valid when

\Re s>0

,

\Im a>0

or

\Re s>1

,

\Im a=0

. When

s=2

and

e^{2\pi ia}=z

, (25.12.13) becomes (25.12.4). …

… ►

•

Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\operatorname{erf}z$ , $x\in[0,5]$ , $y=0.5(.5)3$ , 7D and 8D, respectively; the real and imaginary parts of $\int_{x}^{\infty}e^{\pm\mathrm{i}t^{2}}\,\mathrm{d}t$ , $(1/\sqrt{\pi})e^{\mp\mathrm{i}(x^{2}+(\pi/4))}\int_{x}^{\infty}e^{\pm\mathrm{i% }t^{2}}\,\mathrm{d}t$ , $x=0(.5)20(1)25$ , 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

…

… ►where

j

denotes a real critical point (36.4.1) or (36.4.2), and

\mu

denotes a critical point with complex

t

or

s, t

, connected with

j

by a steepest-descent path (that is, a path where

\Re\Phi=\mathrm{constant}

) in complex

t

or

(s,t)

space. ►In the following subsections, only Stokes sets involving at least one real saddle are included unless stated otherwise. … ►This part of the Stokes set connects two complex saddles. … ►In Figure 36.5.4 the part of the Stokes surface inside the bifurcation set connects two complex saddles. The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions. …

… ►Given the power moments,

\mu_{n}=\int_{a}^{b}x^{n}\,\mathrm{d}\mu(x)

,

n=0,1,2,\dots

, can these be used to find a unique

\mu(x)

, a non-decreasing, real, function of

x

, in the case that the moment problem is determined? Should a unique solution not exist the moment problem is then indeterminant. … ►In what follows we consider only the simple, illustrative, case that

\mu(x)

is continuously differentiable so that

\,\mathrm{d}\mu(x)=w(x)\,\mathrm{d}x

, with

w(x)

real, positive, and continuous on a real interval

[a,b].

The strategy will be to: 1) use the moments to determine the recursion coefficients

\alpha_{n},\beta_{n}

of equations (18.2.11_5) and (18.2.11_8); then, 2) to construct the quadrature abscissas

x_{i}

and weights (or Christoffel numbers)

w_{i}

from the J-matrix of §3.5(vi), equations (3.5.31) and(3.5.32). … ►It is now necessary to take the limit

\varepsilon\to 0{+}

of

F(x^{\prime}+\mathrm{i}\varepsilon)

, and the imaginary part is the required Stieltjes–Perron inversion: …The question is then: how is this possible given only

F_{N}(z)

, rather than

F(z)

itself?

F_{N}(z)

often converges to smooth results for

z

off the real axis for

\Im{z}

at a distance greater than the pole spacing of the

x_{n}

, this may then be followed by approximate numerical analytic continuation via fitting to lower order continued fractions (either Padé, see §3.11(iv), or pointwise continued fraction approximants, see Schlessinger (1968, Appendix)), to

F_{N}(z)

and evaluating these on the real axis in regions of higher pole density that those of the approximating function. Results of low (

~{}2

to

3

decimal digits) precision for

w(x)

are easily obtained for

N\sim 10

to

20

. …

real%20and%20imaginary%20parts

1—10 of 19 matching pages

1: 22.3 Graphics

2: 10.75 Tables

3: 20.10 Integrals

4: 6.19 Tables

5: Software Index

6: 9.18 Tables

7: 25.12 Polylogarithms

8: 7.23 Tables

9: 36.5 Stokes Sets

10: 18.40 Methods of Computation