# 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.

# Examples

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## Basic Examples(5)

Evaluate numerically:

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(32)

### Numerical Evaluation(5)

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:

### Specific Values(4)

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:

### Visualization(4)

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 :

### 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:

### Differentiation(3)

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:

### Integration(3)

Compute the indefinite integral using Integrate:

Definite integral:

More integrals:

### 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)

Use FunctionExpand to convert to Bessel functions:

Integrate expressions with HankelH1:

## Possible Issues(1)

HankelH1 does not automatically evaluate symbolically for half-integer arguments:

Use FunctionExpand to obtain an expanded form: