Fungrim entry: 457aaa

Symbol: AnalyticContinuation —

\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)

— Analytic continuation

Represents the value (or limiting value)

g(b)

where

g(z)

is the unique analytic continuation along the path from

a

b

for the function initially represented by

f(z)

. It is assumed that the expression

f(z)

represents a holomorphic function of

z

in a neighborhood of the initial point

a

. More generally,

a

is allowed to be a pole, branch point or even an essential singularity as long as

f(z)

is holomorphic locally in a cone around the path radiating from

a

. Infinite endpoints are allowed, with the obvious interpretation. Analytic continuation paths are allowed to pass through (isolated) poles of the analytically continued function. The path is not allowed to pass through intermediate branch points, but may end at a branch point.

►AnalyticContinuation(f(z), For(z, a, b)) —

\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)

— Represents the analytic continuation of

f(z)

along the straight-line path from

a

b

►AnalyticContinuation(f(z), For(z, P)) —

\mathop{\text{Continuation}}\limits_{\displaystyle{z: P}} \, f(z)

— Represents the analytic continuation of

f(z)

along the path object

P

►AnalyticContinuation(f(z), For(z, Path(a, b, c))) —

\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b \rightsquigarrow c}} \, f(z)

— Represents the analytic continuation of

f(z)

along the straight-line path

a \rightsquigarrow b \rightsquigarrow c

►AnalyticContinuation(f(z), For(z, CurvePath(Exp(Mul(ConstI, t)), For(t, 0, theta)))) —

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

— Represents the analytic continuation of

f(z)

along the circular path starting at

z = 1

and rotating counterclockwise by the phase

\theta

Definitions:

Fungrim symbol	Notation	Short description
`AnalyticContinuation`	$\mathop{\text{Continuation}}\limits_{\displaystyle{z: a \rightsquigarrow b}} \, f(z)$	Analytic continuation
`Path`	$a \rightsquigarrow b \rightsquigarrow c$	Line path
`CurvePath`	$\left(f(t),\, t : a \rightsquigarrow b\right)$	Path along a curve
`Exp`	${e}^{z}$	Exponential function
`ConstI`	$i$	Imaginary unit

Source code for this entry:

Entry(ID("457aaa"),
    SymbolDefinition(AnalyticContinuation, AnalyticContinuation(f(z), For(z, a, b)), "Analytic continuation"),
    Description("Represents the value (or limiting value)", g(b), "where", g(z), "is the unique analytic continuation along the path from", a, "to", b, "for the function initially represented by", f(z), ". ", "It is assumed that the expression", f(z), "represents a holomorphic function of", z, "in a neighborhood of the initial point", a, ". ", "More generally, ", a, "is allowed to be a pole, branch point or even an essential singularity as long as", f(z), "is holomorphic locally in a cone around", "the path radiating from", a, ". ", "Infinite endpoints are allowed, with the obvious interpretation. Analytic continuation paths are allowed to pass through (isolated) poles of the analytically continued function. ", "The path is not allowed to pass through intermediate branch points, but may end at a branch point."),
    CodeExample(AnalyticContinuation(f(z), For(z, a, b)), "Represents the analytic continuation of", f(z), "along the straight-line path from", a, "to", b, "."),
    CodeExample(AnalyticContinuation(f(z), For(z, P)), "Represents the analytic continuation of", f(z), "along the path object", P, "."),
    CodeExample(AnalyticContinuation(f(z), For(z, Path(a, b, c))), "Represents the analytic continuation of", f(z), "along the straight-line path", Path(a, b, c), "."),
    CodeExample(AnalyticContinuation(f(z), For(z, CurvePath(Exp(Mul(ConstI, t)), For(t, 0, theta)))), "Represents the analytic continuation of", f(z), "along the circular path starting at", Equal(z, 1), "and rotating ", "counterclockwise by the phase", theta, "."))

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