# 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:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix BesselK function using MatrixFunction:

### 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(3)

Specific heat of the relativistic ideal gas per particle:

Find the ultrarelativistic limit:

PDF of geometric mean of two independent exponential random variables:

Surface tension of an electrolyte solution as a function of concentration y:

Onsager law for small concentrations:

## 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:

## Neat Examples(1)

Plot the Riemann surface of :

Plot the Riemann surface of :

Wolfram Research (1988), BesselK, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselK.html (updated 2022).

#### Text

Wolfram Research (1988), BesselK, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselK.html (updated 2022).

#### CMS

Wolfram Language. 1988. "BesselK." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2022. https://reference.wolfram.com/language/ref/BesselK.html.

#### APA

Wolfram Language. (1988). BesselK. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BesselK.html

#### BibTeX

@misc{reference.wolfram_2024_besselk, author="Wolfram Research", title="{BesselK}", year="2022", howpublished="\url{https://reference.wolfram.com/language/ref/BesselK.html}", note=[Accessed: 14-September-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_besselk, organization={Wolfram Research}, title={BesselK}, year={2022}, url={https://reference.wolfram.com/language/ref/BesselK.html}, note=[Accessed: 14-September-2024 ]}