# Fungrim entry: c1bee1

$\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, \log(z) = \log\!\left(-b\right) - \pi i$
Assumptions:$a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Im}(a) < 0 \,\mathbin{\operatorname{and}}\, \operatorname{Im}(b) > 0 \,\mathbin{\operatorname{and}}\, \operatorname{Re}(a) \operatorname{Im}(b) - \operatorname{Re}(b) \operatorname{Im}(a) < 0$
TeX:
\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
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
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
CC$\mathbb{C}$ Complex numbers
Im$\operatorname{Im}(z)$ Imaginary part
Re$\operatorname{Re}(z)$ Real part
Source code for this entry:
Entry(ID("c1bee1"),
Formula(Equal(AnalyticContinuation(Log(z), For(z, a, b)), Sub(Log(Neg(b)), Mul(Pi, ConstI)))),
Variables(a, b),
Assumptions(And(Element(a, CC), Element(b, CC), Less(Im(a), 0), Greater(Im(b), 0), Less(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.

2020-01-31 18:09:28.494564 UTC