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1: 23.16 Graphics

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Figure 23.16.1: Modular functions $\lambda \left(\mathrm{i}y\right)$, $J\left(\mathrm{i}y\right)$, $\eta \left(\mathrm{i}y\right)$ for $0\le y\le 3$. … Magnify

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Figure 23.16.2: Elliptic modular function $\lambda \left(x+\mathrm{i}y\right)$ for $-0.25\le x\le 0.25$, $0.005\le y\le 0.1$. Magnify 3D Help

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Figure 23.16.3: Dedekind’s eta function $\eta \left(x+\mathrm{i}y\right)$ for $-0.0625\le x\le 0.0625$, $0.0001\le y\le 0.07$. Magnify 3D Help

2: 31 Heun Functions

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3: 4.15 Graphics

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Figure 4.15.8: $\sin \left(x+\mathrm{i}y\right)$. Magnify 3D Help

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Figure 4.15.10: $\tan \left(x+\mathrm{i}y\right)$. Magnify 3D Help

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Figure 4.15.11: $\arctan \left(x+\mathrm{i}y\right)$ (principal value). There are branch cuts along the imaginary axis from $-\mathrm{i}\mathrm{\infty }$ to $-\mathrm{i}$ and $\mathrm{i}$ to $\mathrm{i}\mathrm{\infty }$. Magnify 3D Help

… ►The corresponding surfaces for $\cos \left(x+\mathrm{i}y\right)$, $\cot \left(x+\mathrm{i}y\right)$, and $\sec \left(x+\mathrm{i}y\right)$ are similar. … ►The corresponding surfaces for $\arccos \left(x+\mathrm{i}y\right)$, $\mathrm{arccot}\left(x+\mathrm{i}y\right)$, $\mathrm{arcsec}\left(x+\mathrm{i}y\right)$ can be visualized from Figures 4.15.9, 4.15.11, 4.15.13 with the aid of equations (4.23.16)–(4.23.18).

4: 6.5 Further Interrelations

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6.5.1 $$E_1\left(-x\pm \mathrm{i}0\right)=-\mathrm{Ei}\left(x\right)\mp \mathrm{i}\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $x$: real variable

A&S Ref:

5.1.7

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.2 $$\mathrm{Ei}\left(x\right)=-\frac{1}{2}(E_1\left(-x+\mathrm{i}0\right)+E_1\left(-x-\mathrm{i}0\right)),$$

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Symbols:

$E_1\left(z\right)$: exponential integral, $\mathrm{Ei}\left(x\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $x$: real variable

A&S Ref:

5.1.7

Referenced by:

§6.5

Permalink:

http://dlmf.nist.gov/6.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.5 $$\mathrm{Si}\left(z\right)=\frac{1}{2}\mathrm{i}(E_1\left(-\mathrm{i}z\right)-E_1\left(\mathrm{i}z\right))+\frac{1}{2}\pi ,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{Si}\left(z\right)$: sine integral and $z$: complex variable

A&S Ref:

5.2.21

Referenced by:

§6.12(ii), §6.5

Permalink:

http://dlmf.nist.gov/6.5.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.6 $$\mathrm{Ci}\left(z\right)=-\frac{1}{2}(E_1\left(\mathrm{i}z\right)+E_1\left(-\mathrm{i}z\right)),$$

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Symbols:

$\mathrm{Ci}\left(z\right)$: cosine integral, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

5.2.23

Referenced by:

§6.12(ii), §6.5

Permalink:

http://dlmf.nist.gov/6.5.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

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6.5.7 $$\mathrm{g}\left(z\right)\pm \mathrm{i}\mathrm{f}\left(z\right)=E_1\left(\mp \mathrm{i}z\right){\mathrm{e}}^{\mp \mathrm{i}z}.$$

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Symbols:

$\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.11, §6.5, §6.7(iii)

Permalink:

http://dlmf.nist.gov/6.5.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.5 and Ch.6

5: 4.3 Graphics

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	A	B	C	$\overline{\mathrm{C}}$	D	$\overline{\mathrm{D}}$	E	$\overline{\mathrm{E}}$	F
$z$	0	$r$	$r+\mathrm{i}\pi$	$r-\mathrm{i}\pi$	$\mathrm{i}\pi$	$-\mathrm{i}\pi$	$-r+\mathrm{i}\pi$	$-r-\mathrm{i}\pi$	$-r$
$w$	1	${\mathrm{e}}^r$	$-{\mathrm{e}}^r+\mathrm{i}0$	$-{\mathrm{e}}^r-\mathrm{i}0$	$-1+\mathrm{i}0$	$-1-\mathrm{i}0$	$-{\mathrm{e}}^{-r}+\mathrm{i}0$	$-{\mathrm{e}}^{-r}-\mathrm{i}0$	${\mathrm{e}}^{-r}$

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Figure 4.3.2: Conformal mapping of exponential and logarithm. … Magnify

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Figure 4.3.3: $\ln \left(x+\mathrm{i}y\right)$ (principal value). … Magnify 3D Help

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Figure 4.3.4: ${\mathrm{e}}^{x+\mathrm{i}y}$. Magnify 3D Help

6: 4.31 Special Values and Limits

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Table 4.31.1: Hyperbolic functions: values at multiples of $\frac{1}{2}\pi \mathrm{i}$.

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$z$	0	$\frac{1}{2}\pi \mathrm{i}$	$\pi \mathrm{i}$	$\frac{3}{2}\pi \mathrm{i}$	$\mathrm{\infty }$
$\sinh z$	$0$	$\mathrm{i}$	$0$	$-\mathrm{i}$	$\mathrm{\infty }$
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$\tanh z$	$0$	$\mathrm{\infty }\mathrm{i}$	$0$	$-\mathrm{\infty }\mathrm{i}$	$1$
$\mathrm{csch}z$	$\mathrm{\infty }$	$-\mathrm{i}$	$\mathrm{\infty }$	$\mathrm{i}$	$0$
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7: 20.8 Watson’s Expansions

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20.8.1 $$\frac{{\theta }_2(0,q){\theta }_3(z,q){\theta }_4(z,q)}{{\theta }_2(z,q)}=2\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{(-1)}^n{q}^{n^2}{\mathrm{e}}^{\mathrm{i}2nz}}{q^{-n}{\mathrm{e}}^{-\mathrm{i}z}+q^n{\mathrm{e}}^{\mathrm{i}z}}.$$

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Symbols:

${\theta }_j(z,q)$: theta function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $n$: integer, $z$: complex and $q$: nome

Permalink:

http://dlmf.nist.gov/20.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §20.8 and Ch.20

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8: 4.28 Definitions and Periodicity

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4.28.8 $$\sin \left(\mathrm{i}z\right)=\mathrm{i}\sinh z,$$

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Symbols:

$\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\sin z$: sine function and $z$: complex variable

A&S Ref:

4.5.7

Referenced by:

§22.10(ii), §4.29(ii), §4.29(ii), §4.33

Permalink:

http://dlmf.nist.gov/4.28.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §4.28, §4.28 and Ch.4

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4.28.10 $$\tan \left(\mathrm{i}z\right)=\mathrm{i}\tanh z,$$

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Symbols:

$\tanh z$: hyperbolic tangent function, $\mathrm{i}$: imaginary unit, $\tan z$: tangent function and $z$: complex variable

A&S Ref:

4.5.9

Referenced by:

§22.10(ii)

Permalink:

http://dlmf.nist.gov/4.28.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §4.28, §4.28 and Ch.4

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4.28.11 $$\csc \left(\mathrm{i}z\right)=-\mathrm{i}\mathrm{csch}z,$$

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Symbols:

$\csc z$: cosecant function, $\mathrm{csch}z$: hyperbolic cosecant function, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

4.5.10

Permalink:

http://dlmf.nist.gov/4.28.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §4.28, §4.28 and Ch.4

… ►The functions $\sinh z$ and $\cosh z$ have period $2\pi \mathrm{i}$, and $\tanh z$ has period $\pi \mathrm{i}$. The zeros of $\sinh z$ and $\cosh z$ are $z=\mathrm{i}k\pi$ and $z=\mathrm{i}\left(k+\frac{1}{2}\right)\pi$, respectively, $k\in \mathbb{Z}$.

9: 6.4 Analytic Continuation

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6.4.2 $$E_1\left(z{\mathrm{e}}^{2m\pi \mathrm{i}}\right)=E_1\left(z\right)-2m\pi \mathrm{i},$$ $m\in \mathbb{Z}$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers and $z$: complex variable

Permalink:

http://dlmf.nist.gov/6.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.4 $$\mathrm{Ci}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Ci}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(z\right)$: cosine integral, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.5 $$\mathrm{Chi}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pm \pi \mathrm{i}+\mathrm{Chi}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{Chi}\left(z\right)$: hyperbolic cosine integral, $\mathrm{i}$: imaginary unit and $z$: complex variable

Referenced by:

§6.4

Permalink:

http://dlmf.nist.gov/6.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.6 $$\mathrm{f}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\pi {\mathrm{e}}^{\mp \mathrm{i}z}-\mathrm{f}\left(z\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.12(ii), §6.4

Permalink:

http://dlmf.nist.gov/6.4.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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6.4.7 $$\mathrm{g}\left(z{\mathrm{e}}^{\pm \pi \mathrm{i}}\right)=\mp \pi \mathrm{i}{\mathrm{e}}^{\mp \mathrm{i}z}+\mathrm{g}\left(z\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable

Referenced by:

§6.12(ii), §6.4

Permalink:

http://dlmf.nist.gov/6.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §6.4 and Ch.6

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10: 21.4 Graphics

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21.4.1

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Referenced by:

Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1

Permalink:

http://dlmf.nist.gov/21.4.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.4 and Ch.21

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21.4.2

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §21.4 and Ch.21

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21.4.3

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.4 and Ch.21

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Figure 21.4.2: $\mathrm{\Re }\widehat{\theta }\left(x+\mathrm{i}y,0\right|{𝛀}_1)$, $0\le x\le 1$, $0\le y\le 5$. … Magnify 3D Help

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Figure 21.4.4: A real-valued scaled Riemann theta function: $\widehat{\theta }\left(\mathrm{i}x,\mathrm{i}y\right|{𝛀}_1)$, $0\le x\le 4$, $0\le y\le 4$. … Magnify 3D Help

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