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Fungrim entry: a2189a

Continuationz:(Reit,t:0θ)log(z)=log(R)+θi\mathop{\text{Continuation}}\limits_{\displaystyle{z: \left(R {e}^{i t},\, t : 0 \rightsquigarrow \theta\right)}} \, \log(z) = \log(R) + \theta i
Assumptions:R(0,)  and  θRR \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \theta \in \mathbb{R}
\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}
Fungrim symbol Notation Short description
AnalyticContinuationContinuationz:abf(z)\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z) Analytic continuation
Loglog(z)\log(z) Natural logarithm
CurvePath(f(t),t:ab)\left(f(t),\, t : a \rightsquigarrow b\right) Path along a curve
Expez{e}^{z} Exponential function
ConstIii Imaginary unit
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
RRR\mathbb{R} Real numbers
Source code for this entry:
    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