HankelH1
HankelH1[n,z]
gives the Hankel function of the first kind .
Details

- Mathematical function, suitable for both symbolic and numerical manipulation.
is given by
.
- HankelH1[n,z] has a branch cut discontinuity in the complex z plane running from
to
.
- For certain special arguments, HankelH1 automatically evaluates to exact values.
- HankelH1 can be evaluated to arbitrary numerical precision.
- HankelH1 automatically threads over lists.
- HankelH1 can be used with a CenteredInterval object. »
Examples
open allclose allBasic Examples (5)
Plot the real and imaginary parts of the function:
Plot over a subset of the complexes:
Series expansion at the origin:
Series expansion at Infinity:
Scope (33)
Numerical Evaluation (6)
The precision of the output tracks the precision of the input:
Evaluate efficiently at high precision:
HankelH1 threads elementwise over lists and matrices:
HankelH1 can be used with a CenteredInterval object:
Specific Values (4)
Visualization (4)
Function Properties (7)
Complex domain for is the whole plane except
:
It is not defined as a function from to
:
HankelH1 is a complex linear combination of BesselJ and BesselY:
For integer and arbitrary fixed
,
:
is not an analytic function of
:
HankelH1 is not injective over complexes:
Use FindInstance to find inputs that demonstrate it is not injective:
has both singularities and discontinuities along the negative real axis:
TraditionalForm formatting:
Differentiation (3)
Integration (3)
Series Expansions (6)
Find the Taylor expansion using Series:
Plots of the first three approximations around :
General term in the series expansion using SeriesCoefficient:
Asymptotic approximation of HankelH1:
Find series expansion for an arbitrary symbolic direction :
Taylor expansion at a generic point:
HankelH1 can be applied to a power series:
Properties & Relations (2)
Possible Issues (1)
HankelH1 does not automatically evaluate symbolically for half-integer arguments:
Use FunctionExpand to obtain an expanded form:
Text
Wolfram Research (2007), HankelH1, Wolfram Language function, https://reference.wolfram.com/language/ref/HankelH1.html.
CMS
Wolfram Language. 2007. "HankelH1." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/HankelH1.html.
APA
Wolfram Language. (2007). HankelH1. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HankelH1.html