Fungrim home page

Fungrim entry: c1bee1

AnalyticContinuation ⁣(log ⁣(z),z,a,b)=log ⁣(z)πi\operatorname{AnalyticContinuation}\!\left(\log\!\left(z\right), z, a, b\right) = \log\!\left(-z\right) - \pi i
Assumptions:aCandbCandIm ⁣(a)<0andIm ⁣(b)>0andRe ⁣(a)Im ⁣(b)Re ⁣(b)Im ⁣(a)<0a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(a\right) < 0 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(b\right) > 0 \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(a\right) \operatorname{Im}\!\left(b\right) - \operatorname{Re}\!\left(b\right) \operatorname{Im}\!\left(a\right) < 0
TeX:
\operatorname{AnalyticContinuation}\!\left(\log\!\left(z\right), z, a, b\right) = \log\!\left(-z\right) - \pi i

a \in \mathbb{C} \,\mathbin{\operatorname{and}}\, b \in \mathbb{C} \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(a\right) < 0 \,\mathbin{\operatorname{and}}\, \operatorname{Im}\!\left(b\right) > 0 \,\mathbin{\operatorname{and}}\, \operatorname{Re}\!\left(a\right) \operatorname{Im}\!\left(b\right) - \operatorname{Re}\!\left(b\right) \operatorname{Im}\!\left(a\right) < 0
Definitions:
Fungrim symbol Notation Short description
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
ConstPiπ\pi The constant pi (3.14...)
ConstIii Imaginary unit
CCC\mathbb{C} Complex numbers
ImIm ⁣(z)\operatorname{Im}\!\left(z\right) Imaginary part
ReRe ⁣(z)\operatorname{Re}\!\left(z\right) Real part
Source code for this entry:
Entry(ID("c1bee1"),
    Formula(Equal(AnalyticContinuation(Log(z), z, a, b), Sub(Log(Neg(z)), Mul(ConstPi, 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.

2019-09-19 20:12:49.583742 UTC