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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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.

14: 3.4 Differentiation

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B_{-1}^{6}=-\tfrac{1}{48}(36-72t-39t^{2}+52t^{3}+5t^{4}-6t^{5}),

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B_{1}^{6}=\tfrac{1}{48}(36+72t-39t^{2}-52t^{3}+5t^{4}+6t^{5}),

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B_{-3}^{7}=-\tfrac{1}{5040}(48-56t-168t^{2}+140t^{3}+35t^{4}-42t^{5}+7t^{6}),

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B_{-1}^{7}=-\tfrac{1}{240}(144-360t-48t^{2}+260t^{3}-45t^{4}-30t^{5}+7t^{6}),

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B_{3}^{7}=\tfrac{1}{720}(48+8t-192t^{2}-20t^{3}+85t^{4}+6t^{5}-7t^{6}),

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15: 30.9 Asymptotic Approximations and Expansions

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8\beta_{0}=8m^{2}-q^{2}-5,

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2^{10}\beta_{2}=-5(q^{4}+26q^{2}+21)+384m^{2}(q^{2}+1),

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2^{14}\beta_{3}=-33q^{5}-1594q^{3}-5621q+128m^{2}(37q^{3}+167q)-2048m^{4}q.

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2^{20}\beta_{5}=-527q^{7}-61529q^{5}-10\;43961q^{3}-22\;41599q+32m^{2}(5739q^{% 5}+1\;27550q^{3}+2\;98951q)-2048m^{4}(355q^{3}+1505q)+65536m^{6}q.

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2^{13}c_{5}=-527q^{7}-4139q^{5}-5221q^{3}-1009q+m^{2}(939q^{5}+3750q^{3}+1591q% )-m^{4}(465q^{3}+635q)+53m^{6}q.

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16: 4.11 Sums

… ►For infinite series involving logarithms and/or exponentials, see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §44), and Prudnikov et al. (1986a, Chapter 5).

17: 4.27 Sums

… ►For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).

18: 4.24 Inverse Trigonometric Functions: Further Properties

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4.24.1

\operatorname{arcsin}z=z+\frac{1}{2}\frac{z^{3}}{3}+\frac{1\cdot 3}{2\cdot 4}% \frac{z^{5}}{5}+\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}\frac{z^{7}}{7}+\cdots,

|z|\leq 1

ⓘ

Symbols:: $\operatorname{arcsin}\NVar{z}$ : arcsine function and $z$ : complex variable
A&S Ref:: 4.4.40 (where the constraint is $|z|<1$ .)
Permalink:: http://dlmf.nist.gov/4.24.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(i), §4.24 and Ch.4

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4.24.2

\operatorname{arccos}z=(2(1-z))^{1/2}\*\left(1+\sum_{n=1}^{\infty}\frac{1\cdot 3% \cdot 5\cdots(2n-1)}{2^{2n}(2n+1)n!}(1-z)^{n}\right),

|1-z|\leq 2

ⓘ

Symbols:: $!$ : factorial (as in $n!$ ), $\operatorname{arccos}\NVar{z}$ : arccosine function, $n$ : integer and $z$ : complex variable
A&S Ref:: 4.4.41 (which is stated differently and the constraint on $z$ is more restrictive.)
Permalink:: http://dlmf.nist.gov/4.24.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(i), §4.24 and Ch.4

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4.24.3

\operatorname{arctan}z=z-\frac{z^{3}}{3}+\frac{z^{5}}{5}-\frac{z^{7}}{7}+\cdots,

\left|z\right|\leq 1

z\neq\pm\mathrm{i}

ⓘ

Symbols:: $\mathrm{i}$ : imaginary unit, $\operatorname{arctan}\NVar{z}$ : arctangent function and $z$ : complex variable
A&S Ref:: 4.4.42
Referenced by:: §3.10(ii), §4.45(i), §4.45(i)
Permalink:: http://dlmf.nist.gov/4.24.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(i), §4.24 and Ch.4

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4.24.4

\operatorname{arctan}z=\pm\frac{\pi}{2}-\frac{1}{z}+\frac{1}{3z^{3}}-\frac{1}{% 5z^{5}}+\cdots,

\Re z\gtrless 0

|z|\geq 1

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\Re$ : real part and $z$ : complex variable
A&S Ref:: 4.4.42 (has an error. $\frac{\pi}{2}$ should have a negative sign when $\Re z<0$ .)
Referenced by:: §4.45(i)
Permalink:: http://dlmf.nist.gov/4.24.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(i), §4.24 and Ch.4

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4.24.5

\operatorname{arctan}z=\frac{z}{z^{2}+1}\*\left(1+\frac{2}{3}\frac{z^{2}}{1+z^% {2}}+\frac{2\cdot 4}{3\cdot 5}\left(\frac{z^{2}}{1+z^{2}}\right)^{2}+\cdots% \right),

\Re\left(z^{2}\right)>-\tfrac{1}{2}

ⓘ

Symbols:: $\operatorname{arctan}\NVar{z}$ : arctangent function, $\Re$ : real part and $z$ : complex variable
A&S Ref:: 4.4.42 (has an error in the conditions on $z$ .)
Permalink:: http://dlmf.nist.gov/4.24.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §4.24(i), §4.24 and Ch.4

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19: 11.14 Tables

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•

Abramowitz and Stegun (1964, Chapter 12) tabulates $\mathbf{H}_{n}\left(x\right)$ , $\mathbf{H}_{n}\left(x\right)-Y_{n}\left(x\right)$ , and $I_{n}\left(x\right)-\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.1)5$ , $x^{-1}=0(.01)0.2$ to 6D or 7D.

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Agrest et al. (1982) tabulates $\mathbf{H}_{n}\left(x\right)$ and $e^{-x}\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0(.001)5(.005)15(.01)100$ to 11D.

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Barrett (1964) tabulates $\mathbf{L}_{n}\left(x\right)$ for $n=0,1$ and $x=0.2(.005)4(.05)10(.1)19.2$ to 5 or 6S, $x=6(.25)59.5(.5)100$ to 2S.

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•

Abramowitz and Stegun (1964, Chapter 12) tabulates $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t$ and $(2/\pi)\int_{x}^{\infty}t^{-1}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.1)5$ to 5D or 7D; $\int_{0}^{x}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , $\int_{0}^{x}(I_{0}\left(t\right)-\mathbf{L}_{0}\left(t\right))\,\mathrm{d}t-(2% /\pi)\ln x$ , and $\int_{x}^{\infty}t^{-1}(\mathbf{H}_{0}\left(t\right)-Y_{0}\left(t\right))\,% \mathrm{d}t$ for $x^{-1}=0(.01)0.2$ to 6D.

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Agrest et al. (1982) tabulates $\int_{0}^{x}\mathbf{H}_{0}\left(t\right)\,\mathrm{d}t$ and $e^{-x}\int_{0}^{x}\mathbf{L}_{0}\left(t\right)\,\mathrm{d}t$ for $x=0(.001)5(.005)15(.01)100$ to 11D.

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20: 12.19 Tables

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Abramowitz and Stegun (1964, Chapter 19) includes $U\left(a,x\right)$ and $V\left(a,x\right)$ for $\pm a=0(.1)1(.5)5$ , $x=0(.1)5$ , 5S; $W\left(a,\pm x\right)$ for $\pm a=0(.1)1(1)5$ , $x=0(.1)5$ , 4-5D or 4-5S.

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Kireyeva and Karpov (1961) includes $D_{p}\left(x(1+i)\right)$ for $\pm x=0(.1)5$ , $p=0(.1)2$ , and $\pm x=5(.01)10$ , $p=0(.5)2$ , 7D.

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Karpov and Čistova (1964) includes $D_{p}\left(x\right)$ for $p=-2(.1)0$ , $\pm x=0(.01)5$ ; $p=-2(.05)0$ , $\pm x=5(.01)10$ , 6D.

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Karpov and Čistova (1968) includes $e^{-\frac{1}{4}x^{2}}D_{p}\left(-x\right)$ and $e^{-\frac{1}{4}x^{2}}D_{p}\left(ix\right)$ for $x=0(.01)5$ and $x^{-1}$ = 0(.001 or .0001)5, $p=-1(.1)1$ , 7D or 8S.

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•

Zhang and Jin (1996, pp. 455–473) includes $U\left(\pm n-\frac{1}{2},x\right)$ , $V\left(\pm n-\frac{1}{2},x\right)$ , $U\left(\pm\nu-\frac{1}{2},x\right)$ , $V\left(\pm\nu-\frac{1}{2},x\right)$ , and derivatives, $\nu=n+\frac{1}{2}$ , $n=0(1)10(10)30$ , $x=0.5,1,5,10,30,50$ , 8S; $W\left(a,\pm x\right)$ , $W\left(-a,\pm x\right)$ , and derivatives, $a=h(1)5+h$ , $x=0.5,1$ and $a=h(1)5+h$ , $x=5$ , $h=0,0.5$ , 8S. Also, first zeros of $U\left(a,x\right)$ , $V\left(a,x\right)$ , and of derivatives, $a=-6(.5){-1}$ , 6D; first three zeros of $W\left(a,-x\right)$ and of derivative, $a=0(.5)4$ , 6D; first three zeros of $W\left(-a,\pm x\right)$ and of derivative, $a=0.5(.5)5.5$ , 6D; real and imaginary parts of $U\left(a,z\right)$ , $a=-1.5(1)1.5$ , $z=x+iy$ , $x=0.5,1,5,10$ , $y=0(.5)10$ , 8S.

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