real and imaginary parts

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1: 19.3 Graphics

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Figure 19.3.7: $K\left(k\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. … Magnify 3D Help

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Figure 19.3.8: $E\left(k\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. … Magnify 3D Help

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Figure 19.3.9: $\mathrm{\Re }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. … Magnify 3D Help

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Figure 19.3.10: $\mathrm{\Im }\left(K\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. … Magnify 3D Help

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Figure 19.3.12: $\mathrm{\Im }\left(E\left(k\right)\right)$ as a function of complex $k^2$ for $-2\le \mathrm{\Re }\left(k^2\right)\le 2$, $-2\le \mathrm{\Im }\left(k^2\right)\le 2$. … Magnify 3D Help

2: 29.16 Asymptotic Expansions

… ►The approximations for Lamé polynomials hold uniformly on the rectangle $0\le \mathrm{\Re }z\le K$, $0\le \mathrm{\Im }z\le K^{\prime }$, when $nk$ and $n{k}^{\prime }$ assume large real values. …

3: 7.9 Continued Fractions

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7.9.1 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{z}{z^2+}\frac{\frac{1}{2}}{1+}\frac{1}{z^2+}\frac{\frac{3}{2}}{1+}\frac{2}{z^2+}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

A&S Ref:

7.1.14 (in different form)

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

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7.9.2 $$\sqrt{\pi }{\mathrm{e}}^{z^2}\mathrm{erfc}z=\frac{2z}{2z^2+1-}\frac{1\cdot 2}{2z^2+5-}\frac{3\cdot 4}{2z^2+9-}\mathrm{\cdots },$$ $\mathrm{\Re }z>0$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Re }$: real part and $z$: complex variable

Referenced by:

§7.9

Permalink:

http://dlmf.nist.gov/7.9.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

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7.9.3 $$w\left(z\right)=\frac{\mathrm{i}}{\sqrt{\pi }}\frac{1}{z-}\frac{\frac{1}{2}}{z-}\frac{1}{z-}\frac{\frac{3}{2}}{z-}\frac{2}{z-}\mathrm{\cdots },$$ $\mathrm{\Im }z>0$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $z$: complex variable

A&S Ref:

7.1.15 (in different form)

Permalink:

http://dlmf.nist.gov/7.9.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §7.9 and Ch.7

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4: 28.25 Asymptotic Expansions for Large $\mathrm{\Re }z$

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28.25.4 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le 2\pi -\delta$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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28.25.5 $$\mathrm{\Re }z\to +\mathrm{\infty },$$ $-2\pi +\delta \le \mathrm{ph}h+\mathrm{\Im }z\le \pi -\delta$,

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $h$: parameter, $z$: complex variable and $\delta$: arbitrary small positive constant

Permalink:

http://dlmf.nist.gov/28.25.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.25 and Ch.28

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5: 19.32 Conformal Map onto a Rectangle

… ►with $x_1,x_2,x_3$ real constants, has differential ►

19.32.2 $$dz=-\frac{1}{2}\left(\prod _{j=1}^3{(p-x_j)}^{-1/2}\right)dp,$$ $\mathrm{\Im }p>0$; , $j=1,2,3$.

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase and $z(p)$: function

Referenced by:

§19.32

Permalink:

http://dlmf.nist.gov/19.32.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.32 and Ch.19

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$$z(x_3)=R_F(x_3-x_1,x_3-x_2,0)=-\mathrm{i}{R}_F(0,x_1-x_3,x_2-x_3).$$

►As $p$ proceeds along the entire real axis with the upper half-plane on the right, $z$ describes the rectangle in the clockwise direction; hence $z(x_3)$ is negative imaginary. …

6: 23.1 Special Notation

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$𝕃$	lattice in $ℂ$.
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$z=x+\mathrm{i}y$	complex variable, except in §§23.20(ii), 23.21(iii).
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7: 27.14 Unrestricted Partitions

… ►The corresponding unrestricted partition function is denoted by $p\left(n\right)$, and the summands are called parts; see §26.9(i). … ►The number of partitions of $n$ into at most $k$ parts is denoted by $p_k\left(n\right)$; again see §26.9(i). … ►

27.14.14 $$\eta \left(\frac{a\tau +b}{c\tau +d}\right)=\epsilon {(-\mathrm{i}(c\tau +d))}^{\frac{1}{2}}\eta \left(\tau \right),$$

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Symbols:

$\eta \left(\tau \right)$: Dedekind’s eta function (or Dedekind modular function), $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit, $a$: integer, $b$: integer, $c$: integer and $d$: integer

Referenced by:

§17.2(i)

Permalink:

http://dlmf.nist.gov/27.14.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §27.14(iv), §27.14 and Ch.27

►where $\epsilon =\exp \left(\pi \mathrm{i}(((a+d)/(12c))-s(d,c))\right)$ and $s(d,c)$ is given by (27.14.11). … ►The 24th power of $\eta \left(\tau \right)$ in (27.14.12) with ${\mathrm{e}}^{2\pi \mathrm{i}\tau }=x$ is an infinite product that generates a power series in $x$ with integer coefficients called Ramanujan’s tau function $\tau \left(n\right)$: …

8: 23.11 Integral Representations

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9: 5.4 Special Values and Extrema

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5.4.16 $$\mathrm{\Im }\psi \left(\mathrm{i}y\right)=\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.11

Permalink:

http://dlmf.nist.gov/5.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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5.4.17 $$\mathrm{\Im }\psi \left(\frac{1}{2}+\mathrm{i}y\right)=\frac{\pi }{2}\tanh \left(\pi y\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\tanh z$: hyperbolic tangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.12

Permalink:

http://dlmf.nist.gov/5.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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5.4.18 $$\mathrm{\Im }\psi \left(1+\mathrm{i}y\right)=-\frac{1}{2y}+\frac{\pi }{2}\coth \left(\pi y\right).$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\coth z$: hyperbolic cotangent function, $\mathrm{\Im }$: imaginary part, $\mathrm{i}$: imaginary unit and $y$: real variable

A&S Ref:

6.3.13

Referenced by:

§5.4(ii)

Permalink:

http://dlmf.nist.gov/5.4.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §5.4(ii), §5.4 and Ch.5

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10: 4.2 Definitions

… ►The real and imaginary parts of $\ln z$ are given by … ►

4.2.29 $$|z^a|={|z|}^{\mathrm{\Re }a}\exp \left(-(\mathrm{\Im }a)\mathrm{ph}z\right),$$

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Symbols:

$\exp z$: exponential function, $\mathrm{\Im }$: imaginary part, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.9

Permalink:

http://dlmf.nist.gov/4.2.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

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4.2.30 $$\mathrm{ph}\left(z^a\right)=(\mathrm{\Re }a)\mathrm{ph}z+(\mathrm{\Im }a)\ln |z|,$$

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Symbols:

$\mathrm{\Im }$: imaginary part, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $a$: real or complex constant and $z$: complex variable

A&S Ref:

4.2.10

Permalink:

http://dlmf.nist.gov/4.2.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

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4.2.36 $$-\pi \le \mathrm{\Im }\left(\frac{1}{a}\mathrm{Ln}w\right)\le \pi .$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{\Im }$: imaginary part, $\mathrm{Ln}z$: general logarithm function and $a$: real or complex constant

Permalink:

http://dlmf.nist.gov/4.2.E36

Encodings:

pMML, png, TeX

See also:

Annotations for §4.2(iv), §4.2(iv), §4.2 and Ch.4

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