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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: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}(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
AnalyticContinuationContinuationz:abf(z)\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z) Analytic continuation
Loglog(z)\log(z) Natural logarithm
ConstPiπ\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:
Entry(ID("ed6590"),
    Formula(Equal(AnalyticContinuation(Log(z), For(z, a, b)), Add(Log(Neg(b)), Mul(ConstPi, 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))))

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

2019-10-05 13:11:19.856591 UTC