imaginary part

SearchAdvancedHelp

(0.006 seconds)

11—20 of 188 matching pages

11: 4.2 Definitions

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

4.2.23 $$\mathrm{ph}\left(\exp z\right)=\mathrm{\Im }z+2k\pi ,$$ $k\in \mathbb{Z}$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\exp z$: exponential function, $\mathrm{\Im }$: imaginary part, $\mathbb{Z}$: set of all integers, $\mathrm{ph}$: phase, $k$: integer and $z$: complex variable

A&S Ref:

4.2.14 (modified)

Permalink:

http://dlmf.nist.gov/4.2.E23

Encodings:

pMML, png, TeX

See also:

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

… ►

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

ⓘ

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

►

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

ⓘ

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

… ►

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

ⓘ

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

…

12: 10.52 Limiting Forms

… ►

$${𝗃}_n\left(z\right)=z^{-1}\sin \left(z-\frac{1}{2}n\pi \right)+{\mathrm{e}}^{|\mathrm{\Im }z|}O\left(z^{-2}\right),$$

►

$${𝗒}_n\left(z\right)=-z^{-1}\cos \left(z-\frac{1}{2}n\pi \right)+{\mathrm{e}}^{|\mathrm{\Im }z|}O\left(z^{-2}\right),$$

…

13: 4.8 Identities

… ►

4.8.9 $$\ln \left(\exp z\right)=z,$$ $-\pi \le \mathrm{\Im }z\le \pi$,

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\exp z$: exponential function, $\mathrm{\Im }$: imaginary part, $\ln z$: principal branch of logarithm function and $z$: complex variable

A&S Ref:

4.2.3

Permalink:

http://dlmf.nist.gov/4.8.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(i), §4.8 and Ch.4

… ►where the integer $k$ is chosen so that $\mathrm{\Re }\left(-\mathrm{i}z\ln a\right)+2k\pi \in [-\pi ,\pi ]$. … ►

4.8.17 $${({\mathrm{e}}^{z_1})}^{z_2}={\mathrm{e}}^{z_1{z}_2},$$ $-\pi \le \mathrm{\Im }{z}_1\le \pi$.

ⓘ

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part and $z$: complex variable

A&S Ref:

4.2.19

Permalink:

http://dlmf.nist.gov/4.8.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §4.8(ii), §4.8 and Ch.4

…

14: 25.4 Reflection Formulas

… ►

25.4.5 $${(-1)}^k{\zeta }^{(k)}\left(1-s\right)=\frac{2}{{(2\pi )}^s}\sum _{m=0}^k\sum _{r=0}^m\left(\genfrac{}{}{0.0pt}{}{k}{m}\right)\left(\genfrac{}{}{0.0pt}{}{m}{r}\right)\left(\mathrm{\Re }\left(c^{k-m}\right)\cos \left(\frac{1}{2}\pi s\right)+\mathrm{\Im }\left(c^{k-m}\right)\sin \left(\frac{1}{2}\pi s\right)\right){\mathrm{\Gamma }}^{(r)}\left(s\right){\zeta }^{(m-r)}\left(s\right),$$

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{\Im }$: imaginary part, $\mathrm{\Re }$: real part, $\sin z$: sine function, $k$: nonnegative integer, $m$: nonnegative integer, $s$: complex variable and $c$

Keywords:

derivative

Source:

Apostol (1985a, (6), p. 224)

Permalink:

http://dlmf.nist.gov/25.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §25.4 and Ch.25

…

15: 10.14 Inequalities; Monotonicity

… ►

10.14.3 $$|J_n\left(z\right)|\le {\mathrm{e}}^{|\mathrm{\Im }z|},$$ $n\in \mathbb{Z}$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $\mathbb{Z}$: set of all integers, $n$: integer and $z$: complex variable

Permalink:

http://dlmf.nist.gov/10.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14 and Ch.10

►

10.14.4 $$|J_{\nu }\left(z\right)|\le \frac{{|\frac{1}{2}z|}^{\nu }{\mathrm{e}}^{|\mathrm{\Im }z|}}{\mathrm{\Gamma }\left(\nu +1\right)},$$ $\nu \ge -\frac{1}{2}$.

ⓘ

Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: imaginary part, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.1.62 (corrected)

Permalink:

http://dlmf.nist.gov/10.14.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §10.14 and Ch.10

…

16: 23.15 Definitions

… ►In §§23.15–23.19, $k$ and $k^{\prime }$ $(\in ℂ)$ denote the Jacobi modulus and complementary modulus, respectively, and $q={\mathrm{e}}^{\mathrm{i}\pi \tau }$ ($\mathrm{\Im }\tau >0$) denotes the nome; compare §§20.1 and 22.1. … ►A modular function $f(\tau )$ is a function of $\tau$ that is meromorphic in the half-plane $\mathrm{\Im }\tau >0$, and has the property that for all $𝒜\in \text{SL}(2,\mathbb{Z})$, or for all $𝒜$ belonging to a subgroup of SL$(2,\mathbb{Z})$, ►

23.15.5 $$f(𝒜\tau )=c_{𝒜}{(c\tau +d)}^{\mathrm{\ell }}f(\tau ),$$ $\mathrm{\Im }\tau >0$,

ⓘ

Symbols:

$\mathrm{\Im }$: imaginary part, $\tau$: complex variable, $𝒜$: bilinear transformation, $c$: integer, $d$: integer, $f(\tau )$: modular function, $c_{𝒜}$: constant and $\mathrm{\ell }$: level

Permalink:

http://dlmf.nist.gov/23.15.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §23.15(i), §23.15 and Ch.23

…

17: 20.13 Physical Applications

… ►In the singular limit $\mathrm{\Im }\tau \to 0+$, the functions ${\theta }_j\left(z\right|\tau )$, $j=1,2,3,4$, become integral kernels of Feynman path integrals (distribution-valued Green’s functions); see Schulman (1981, pp. 194–195). …

18: 28.9 Zeros

… ►Furthermore, for $q>0$ ${\mathrm{ce}}_m(z,q)$ and ${\mathrm{se}}_m(z,q)$ also have purely imaginary zeros that correspond uniquely to the purely imaginary $z$-zeros of $J_m\left(2\sqrt{q}\cos z\right)$ (§10.21(i)), and they are asymptotically equal as $q\to 0$ and $\left|\mathrm{\Im }z\right|\to \mathrm{\infty }$. There are no zeros within the strip other than those on the real and imaginary axes. …

19: 15.17 Mathematical Applications

… ►The quotient of two solutions of (15.10.1) maps the closed upper half-plane $\mathrm{\Im }z\ge 0$ conformally onto a curvilinear triangle. …

20: 28.17 Stability as $x\to \pm \mathrm{\infty }$

… ►However, if $\mathrm{\Im }\nu \ne 0$, then $(a,q)$ always comprises an unstable pair. …