# SphericalHankelH1

SphericalHankelH1[n,z]

gives the spherical Hankel function of the first kind .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• SphericalHankelH1 is given in terms of ordinary Hankel functions by .
• SphericalHankelH1[n,z] has a branch cut discontinuity in the complex z plane running from to .
• Explicit symbolic forms for integer n can be obtained using FunctionExpand.
• For certain special arguments, SphericalHankelH1 automatically evaluates to exact values.
• SphericalHankelH1 can be evaluated to arbitrary numerical precision.
• SphericalHankelH1 automatically threads over lists.
• SphericalHankelH1 can be used with CenteredInterval objects. »

# Examples

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## Basic Examples(6)

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:

Series expansion at a singular point:

## Scope(31)

### Numerical Evaluation(5)

Evaluate numerically:

Evaluate to high precision:

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

Complex number inputs:

Evaluate efficiently at high precision:

SphericalHankelH1 can be used with CenteredInterval objects:

### Specific Values(4)

Limiting value at infinity:

SphericalHankelH1 for symbolic n:

Find the first positive zero of imaginary part of SphericalHankelH1:

Different SphericalHankelH1 types give different symbolic forms:

### Visualization(3)

Plot the absolute values of SphericalHankelH1 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 :

SphericalHankelH1 is a complex linear combination of SphericalBesselJ and SphericalBesselY: is not an analytic function:

SphericalHankelH1 is not injective over complexes:

Use FindInstance to find inputs that demonstrate it is not injective: has both singularities and discontinuities along the non-positive real axis:

### Differentiation(3)

First derivative with respect to :

Higher derivatives with respect to :

Plot the absolute values of the higher derivatives of with respect to :

Formula for the  derivative with respect to z:

### Integration(3)

Compute the indefinite integral using Integrate:

Definite integral:

More integrals:

### Series Expansions(4)

Find the Taylor expansion using Series:

General term in the series expansion using SeriesCoefficient:

Find the series expansion at Infinity:

Taylor expansion at a generic point:

### Function Identities and Simplifications(2)

Use FullSimplify to simplify spherical Hankel functions of the first kind:

Recurrence relations: