Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number input:

Evaluate efficiently at high precision:

HankelH1 threads elementwise over lists and matrices:

Limiting value at infinity:

Evaluate HankelH1 symbolically for noninteger orders:

HankelH1 for symbolic n:

Find the first positive maximum for the real part of HankelH1[1/2,x]:

Visualize the result:

Plot the absolute values and phase of the HankelH1 function for various orders:

Plot the real and imaginary parts of the HankelH1 function for various orders:

Plot the real part of :

Plot the imaginary part of :

Plot the real part of :

Plot the imaginary part of :

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:

First derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when n=2:

Formula for the derivative with respect to z:

Compute the indefinite integral using Integrate:

Definite integral:

More integrals:

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: