Fungrim entry: a2189a

\mathop{\text{Continuation}}\limits_{\displaystyle{z: \left(R {e}^{i t},\, t : 0 \rightsquigarrow \theta\right)}} \, \log(z) = \log(R) + \theta i

Assumptions:

R \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \theta \in \mathbb{R}

TeX:

\mathop{\text{Continuation}}\limits_{\displaystyle{z: \left(R {e}^{i t},\, t : 0 \rightsquigarrow \theta\right)}} \, \log(z) = \log(R) + \theta i

R \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; \theta \in \mathbb{R}

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
`CurvePath`	$\left(f(t),\, t : a \rightsquigarrow b\right)$	Path along a curve
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit
`OpenInterval`	$\left(a, b\right)$	Open interval
`Infinity`	$\infty$	Positive infinity
`RR`	$\mathbb{R}$	Real numbers

Source code for this entry:

Entry(ID("a2189a"),
    Formula(Equal(AnalyticContinuation(Log(z), For(z, CurvePath(Mul(R, Exp(Mul(ConstI, t))), For(t, 0, theta)))), Add(Log(R), Mul(theta, ConstI)))),
    Variables(R, theta),
    Assumptions(And(Element(R, OpenInterval(0, Infinity)), Element(theta, RR))))

Topics using this entry

Natural logarithm