Fungrim entry: ed6590

\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("ed6590"),
    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

Natural logarithm