Fungrim home page

Fungrim entry: ed6590

Continuationz:ablog(z)=log ⁣(b)+πi\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, \log(z) = \log\!\left(-b\right) + \pi i
Assumptions:aC  and  bC  and  Im(a)>0  and  Im(b)<0  and  Re(a)Im(b)Re(b)Im(a)>0a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Im}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(b) < 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) \operatorname{Im}(b) - \operatorname{Re}(b) \operatorname{Im}(a) > 0
\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, \log(z) = \log\!\left(-b\right) + \pi i

a \in \mathbb{C} \;\mathbin{\operatorname{and}}\; b \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Im}(a) > 0 \;\mathbin{\operatorname{and}}\; \operatorname{Im}(b) < 0 \;\mathbin{\operatorname{and}}\; \operatorname{Re}(a) \operatorname{Im}(b) - \operatorname{Re}(b) \operatorname{Im}(a) > 0
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
Piπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
ImIm(z)\operatorname{Im}(z) Imaginary part
ReRe(z)\operatorname{Re}(z) Real part
Source code for this entry:
    Formula(Equal(AnalyticContinuation(Log(z), For(z, a, b)), Add(Log(Neg(b)), Mul(Pi, ConstI)))),
    Variables(a, b),
    Assumptions(And(Element(a, CC), Element(b, CC), Greater(Im(a), 0), Less(Im(b), 0), Greater(Sub(Mul(Re(a), Im(b)), Mul(Re(b), Im(a))), 0))))

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