# BesselK

BesselK[n,z]

gives the modified Bessel function of the second kind .

# Details • Mathematical function, suitable for both symbolic and numerical manipulation.
• satisfies the differential equation .
• BesselK[n,z] has a branch cut discontinuity in the complex z plane running from to .
• FullSimplify and FunctionExpand include transformation rules for BesselK.
• For certain special arguments, BesselK automatically evaluates to exact values.
• BesselK can be evaluated to arbitrary numerical precision.
• BesselK automatically threads over lists.
• BesselK can be used with Interval and CenteredInterval objects. »

# Examples

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## Basic Examples(5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at Infinity:

## Scope(45)

### Numerical Evaluation(6)

Evaluate numerically:

Evaluate to high precision:

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

Evaluate for complex arguments and parameters:

Evaluate BesselK efficiently at high precision:

BesselK threads elementwise over lists and matrices:

BesselK can be used with Interval and CenteredInterval objects:

### Specific Values(4)

Value of BesselK for integers ( ) orders at :

For half-integer index, BesselK evaluates to elementary functions:

Limiting value at infinity:

Find the value of satisfying equation :

Visualize the result:

### Visualization(3)

Plot the BesselK function for integer orders ( ):

Plot the real and imaginary parts of the BesselK function for integer orders ( ):

Plot the real part of :

Plot the imaginary part of :

### Function Properties(11) is defined for all real values greater than 0:

Complex domain:

For real , achieves all positive real values:

BesselK is an even function with respect to the first parameter: is not an analytic function:

BesselK is neither non-decreasing nor non-increasing: is injective for all real : is not surjective for any real :

BesselK is neither non-negative nor non-positive:

BesselK has both singularity and discontinuity for z0: is convex on its real domain:

### Differentiation(3)

First derivative:

Higher derivatives:

Plot higher derivatives for order :

Formula for the  derivative:

### Integration(3)

Indefinite integral of BesselK:

Integrate expressions involving BesselK:

Definite integral of BesselK over its real domain:

### Series Expansions(5)

Series expansion for around :

Plot the first three approximations for around :

General term in the series expansion of BesselK:

Asymptotic expansion for BesselK:

Taylor expansion at a generic point:

BesselK can be applied to a power series:

### Integral Transforms(3)

Compute the Mellin transform using MellinTransform:

### Function Identities and Simplifications(3)

Use FullSimplify to simplify Bessel functions:

Verify the identity :

Recurrence relations :

### Function Representations(4)

Integral representation of BesselK:

Represent using BesselI and Sin:

BesselK can be represented in terms of MeijerG:

BesselK can be represented as a DifferenceRoot:

## Applications(1)

Specific heat of the relativistic ideal gas per particle:

Find the ultrarelativistic limit:

## Properties & Relations(2)

Use FullSimplify to simplify Bessel functions:

The exponential generating function for BesselK:

## Possible Issues(1)

With numeric arguments, half-integer Bessel functions are not automatically evaluated:

For symbolic arguments they are:

This can lead to inaccuracies in machine-precision evaluation: