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1: Possible Errors in DLMF

… ►One source of confusion, rather than actual errors, are some new functions which differ from those in Abramowitz and Stegun (1964) by scaling, shifts or constraints on the domain; see the Info box (click or hover over the

icon) for links to defining formula. There are also cases where browser bugs or poor fonts can be misleading; you can verify MathML display by comparing the to the images or TeXfound under Encodings in the Info boxes (see About MathML). … ►Nevertheless, it is quite possible that errors remain and we would appreciate knowing about them, so that we can make the necessary corrections. …

2: Need Help?

… ►Alas, sometimes things go wrong; Perhaps some of these topics can help. … ►

Graphics

•

How can I view ‘VRML’? What is it? See Viewing DLMF Interactive 3D Graphics.
•

What do the colors mean? See About Color Map.

… ►

Finding Things

•

How do I search within DLMF? See Guide to Searching the DLMF.
•

See also the Index or Notations sections.
•

Links to definitions, keywords, annotations and other interesting information can be found in the Info boxes by clicking or hovering the mouse over the icon next to each formula, table, figure, and section heading.

3: 10.75 Tables

… ►

•

The main tables in Abramowitz and Stegun (1964, Chapter 9) give $e^{-x}I_{n}\left(x\right)$ , $e^{x}K_{n}\left(x\right)$ , $n=0,1,2$ , $x=0(.1)10(.2)20$ , 8D–10D or 10S; $\sqrt{x}e^{-x}I_{n}\left(x\right)$ , $(\sqrt{x}/\pi)$ $e^{x}K_{n}\left(x\right)$ , $n=0,1,2$ , $1/x=0(.002)0.05$ ; $K_{0}\left(x\right)+I_{0}\left(x\right)\ln x$ , $x(K_{1}\left(x\right)-I_{1}\left(x\right)\ln x)$ , $x=0(.1)2$ , 8D; $e^{-x}I_{n}\left(x\right)$ , $e^{x}K_{n}\left(x\right)$ , $n=3(1)9$ , $x=0(.2)10(.5)20$ , 5S; $I_{n}\left(x\right)$ , $K_{n}\left(x\right)$ , $n=0(1)20(10)50,100$ , $x=1,2,5,10,50,100$ , 9–10S.

►

•

Achenbach (1986) tabulates $I_{0}\left(x\right)$ , $I_{1}\left(x\right)$ , $K_{0}\left(x\right)$ , $K_{1}\left(x\right)$ , $x=0(.1)8$ , 19D or 19–21S.

►

•

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.

… ►

•

The main tables in Abramowitz and Stegun (1964, Chapter 10) give $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ $n=0(1)8$ , $x=0(.1)10$ , 5–8S; $\mathsf{j}_{n}\left(x\right)$ , $\mathsf{y}_{n}\left(x\right)$ $n=0(1)20(10)50$ , 100, $x=1,2,5,10,50,100$ , 10S; ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $n=0,1,2$ , $x=0(.1)5$ , 4–9D; ${\mathsf{i}^{(1)}_{n}}\left(x\right)$ , $\mathsf{k}_{n}\left(x\right)$ , $n=0(1)20(10)50$ , 100, $x=1,2,5,10,50,100$ , 10S. (For the notation see §10.1 and §10.47(ii).)

►

•

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.)

…

4: 19.29 Reduction of General Elliptic Integrals

… ►There are only three distinct

U

’s with subscripts

\leq 4

, and at most one of them can be 0 because the

d

’s are nonzero. … ►(19.2.3) can be written …The integrals in (19.29.4), (19.29.7), and (19.29.8) are

I(\boldsymbol{{0}})

I(\mathbf{e}_{\alpha}-\mathbf{e}_{\delta})

, and

I(\mathbf{e}_{\alpha}-\mathbf{e}_{5})

, respectively. … ►

I(\mathbf{m})

can be reduced to a linear combination of basic integrals and algebraic functions. … ►It can be expressed in terms of symmetric integrals by setting

a_{5}=1

and

b_{5}=0

in (19.29.8). …

5: Customize DLMF

…You can customize the appearance and functionality of the DLMF site using these selections. …

6: 2.5 Mellin Transform Methods

… ►The sum in (2.5.6) is taken over all poles of

x^{-z}\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(1-z\right)\mathscr{M}\mskip-3.% 0muh\mskip 3.0mu\left(z\right)

in the strip

d<\Re z<c

, and it provides the asymptotic expansion of

I(x)

for small values of

x

. Similarly, if

\mathscr{M}\mskip-3.0muf\mskip 3.0mu\left(1-z\right)

and

\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right)

can be continued analytically to meromorphic functions in a right half-plane, and if the vertical line of integration can be translated to the right, then we obtain an asymptotic expansion for

I(x)

for large values of

x

. … ►Hence we can extend the definition of the Mellin transform of

f

by setting … ►The asymptotic expansion of

I(x)

is then obtained from (2.5.29). … ►where …

7: 8.17 Incomplete Beta Functions

… ►Further integral representations can be obtained by combining the results given in §8.17(ii) with §15.6. … ►

8.17.13

(a+b)I_{x}\left(a,b\right)=aI_{x}\left(a+1,b\right)+bI_{x}\left(a,b+1\right),

ⓘ

Symbols:: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : 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

… ►

8.17.16

aI_{x}\left(a+1,b\right)=(a+cx)I_{x}\left(a,b\right)-cxI_{x}\left(a-1,b\right),

ⓘ

Symbols:: $I_{\NVar{x}}\left(\NVar{a},\NVar{b}\right)$ : 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

… ►The

4m

and

4m+1

convergents are less than

I_{x}\left(a,b\right)

, and the

4m+2

and

4m+3

convergents are greater than

I_{x}\left(a,b\right)

. … ►For

x>(a+1)/(a+b+2)

1-x<(b+1)/(a+b+2)

, more rapid convergence is obtained by computing

I_{1-x}\left(b,a\right)

and using (8.17.4). …

8: 8.23 Statistical Applications

… ►The function

\mathrm{B}_{x}\left(a,b\right)

and its normalization

I_{x}\left(a,b\right)

play a similar role in statistics in connection with the beta distribution; see Johnson et al. (1995, pp. 210–275). In queueing theory the Erlang loss function is used, which can be expressed in terms of the reciprocal of

Q\left(a,x\right)

; see Jagerman (1974) and Cooper (1981, pp. 80, 316–319). …

9: Guide to Searching the DLMF

… ►In DLMF, you can search not only for text but also for mathematical expressions. …From there you can also access an advanced search page where you can control certain settings, narrowing the search to certain chapters, or restricting the results to equations, graphs, tables, or bibliographic items. … ►You can use in math queries all the symbols and commands defined in LaTeX (you can omit the

\backslash

), and some additional convenient ones, as well as the special functions’ names: … ►The syntax of the special functions can be LaTeX-like or as employed in widely used computer algebra systems. For example, for the Bessel function

K_{n}\left(z\right)

, you can write K_n(z), BesselK_n(z), BesselK(n,z), or BesselK[n,z]. …

10: 1.4 Calculus of One Variable

… ►If

f(x_{1})\leq f(x_{2})

for every pair

x_{1}

x_{2}

in an interval

I

such that

x_{1}<x_{2}

, then

f(x)

is nondecreasing on

I

. … ►If

f(x)

is continuous on an interval

I

save for a finite number of simple discontinuities, then

f(x)

is piecewise (or sectionally) continuous on

I

. … ►When

n\geq 1

f

is continuously differentiable on

I

. When

n

is unbounded,

f

is infinitely differentiable on

I

and we write

f\in C^{\infty}(I)

. … ►Stieltjes integrability for

f

with respect to

\alpha

can be defined similarly as Riemann integrability in the case that

\alpha(x)

is differentiable with respect to

x

; a generalization follows below. …