Enable JavaScript to interact with content and submit forms on Wolfram websites. Learn how

BesselK

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

open allclose all

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 z≤0:

is convex on its real domain:

TraditionalForm formatting:

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)

LaplaceTransform:

HankelTransform:

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 ultra‐relativistic 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 :

Top