SphericalHankelH1

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`SphericalHankelH1`

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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)Summary of the most common use cases

Evaluate numerically:

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Plot the real and imaginary parts of the function:

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Plot over a subset of the complexes:

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Series expansion at the origin:

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Series expansion at Infinity:

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Series expansion at a singular point:

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Scope (32)Survey of the scope of standard use cases

Numerical Evaluation (6)

Evaluate numerically:

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Evaluate to high precision:

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

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Complex number inputs:

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Evaluate efficiently at high precision:

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SphericalHankelH1 can be used with CenteredInterval objects:

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Compute the elementwise values of an array:

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Or compute the matrix SphericalHankelH1 function using MatrixFunction:

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Specific Values (4)

Limiting value at infinity:

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SphericalHankelH1 for symbolic n:

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Find the first positive zero of imaginary part of SphericalHankelH1:

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Different SphericalHankelH1 types give different symbolic forms:

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Visualization (3)

Plot the absolute values of SphericalHankelH1 function for various orders:

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Plot the real part of :

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Plot the imaginary part of :

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Plot the real part of :

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Plot the imaginary part of :

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Function Properties (7)

Complex domain for is the whole plane except :

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It is not defined as a function from to :

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SphericalHankelH1 is a complex linear combination of SphericalBesselJ and SphericalBesselY:

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SphericalHankelH1 threads elementwise over lists:

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is not an analytic function:

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SphericalHankelH1 is not injective over complexes:

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Use FindInstance to find inputs that demonstrate it is not injective:

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has both singularities and discontinuities along the non-positive real axis:

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TraditionalForm formatting:

Differentiation (3)

First derivative with respect to :

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Higher derivatives with respect to :

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

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Formula for the derivative with respect to z:

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Integration (3)

Compute the indefinite integral using Integrate:

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Definite integral:

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More integrals:

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Series Expansions (4)

Find the Taylor expansion using Series:

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General term in the series expansion using SeriesCoefficient:

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Find the series expansion at Infinity:

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Taylor expansion at a generic point:

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Function Identities and Simplifications (2)

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

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Recurrence relations:

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Properties & Relations (1)Properties of the function, and connections to other functions

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