Fungrim entry: a2189a

$\mathop{\text{Continuation}}\limits_{\displaystyle{z: \left(R {e}^{i t},\, t : 0 \rightsquigarrow \theta\right)}} \, \log(z) = \log(R) + \theta i$
Assumptions:$R \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \theta \in \mathbb{R}$
TeX:
\mathop{\text{Continuation}}\limits_{\displaystyle{z: \left(R {e}^{i t},\, t : 0 \rightsquigarrow \theta\right)}} \, \log(z) = \log(R) + \theta i

R \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \theta \in \mathbb{R}
Definitions:
Fungrim symbol Notation Short description
AnalyticContinuation$\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)$ Analytic continuation
Log$\log(z)$ Natural logarithm
CurvePath$\left(f(t),\, t : a \rightsquigarrow b\right)$ Path along a curve
Exp${e}^{z}$ Exponential function
ConstI$i$ Imaginary unit
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
RR$\mathbb{R}$ Real numbers
Source code for this entry:
Entry(ID("a2189a"),
Formula(Equal(AnalyticContinuation(Log(z), For(z, CurvePath(Mul(R, Exp(Mul(ConstI, t))), For(t, 0, theta)))), Add(Log(R), Mul(theta, ConstI)))),
Variables(R, theta),
Assumptions(And(Element(R, OpenInterval(0, Infinity)), Element(theta, RR))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC