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:

Compute the elementwise values of an array:

Or compute the matrix SphericalHankelH1 function using MatrixFunction:

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:

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 :

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:

SphericalHankelH1 threads elementwise over lists:

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:

TraditionalForm formatting:

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:

Compute the indefinite integral using Integrate:

Definite integral:

More integrals:

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:

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

Recurrence relations: