gives the modified Bessel function of the second kind .


  • 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.


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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  (44)

Numerical Evaluation  (5)

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:

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 TemplateBox[{0, x}, BesselK] satisfying equation TemplateBox[{0, x}, BesselK]=2:

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 TemplateBox[{0, {x, +, {ⅈ,  , y}}}, BesselK]:

Plot the imaginary part of TemplateBox[{0, {x, +, {ⅈ,  , y}}}, BesselK]:

Function Properties  (11)

is defined for all real values greater than 0:

Complex domain:

For real , TemplateBox[{n, x}, BesselK] achieves all positive real values:

BesselK is an even function with respect to the first parameter:

TemplateBox[{n, z}, BesselK] is not an analytic function:

BesselK is neither non-decreasing nor non-increasing:

TemplateBox[{n, z}, BesselK] is injective for all real :

TemplateBox[{n, z}, BesselK] is not surjective for any real :

BesselK is neither non-negative nor non-positive:

BesselK has both singularity and discontinuity for z0:

TemplateBox[{n, z}, BesselK] is convex on its real domain:

TraditionalForm formatting:

Differentiation  (3)

First derivative:

Higher derivatives:

Plot higher derivatives for order :

Formula for the ^(th) 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 TemplateBox[{nu, z}, BesselI] TemplateBox[{{nu, +, 1}, z}, BesselK]+TemplateBox[{{nu, +, 1}, z}, BesselI] TemplateBox[{nu, z}, BesselK]=1/z:

Recurrence relations z (TemplateBox[{{n, +, 1}, z}, BesselK]-TemplateBox[{{n, -, 1}, z}, BesselK])=2 n TemplateBox[{n, z}, BesselK]:

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:

Wolfram Research (1988), BesselK, Wolfram Language function, (updated 2002).


Wolfram Research (1988), BesselK, Wolfram Language function, (updated 2002).


@misc{reference.wolfram_2021_besselk, author="Wolfram Research", title="{BesselK}", year="2002", howpublished="\url{}", note=[Accessed: 21-October-2021 ]}


@online{reference.wolfram_2021_besselk, organization={Wolfram Research}, title={BesselK}, year={2002}, url={}, note=[Accessed: 21-October-2021 ]}


Wolfram Language. 1988. "BesselK." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2002.


Wolfram Language. (1988). BesselK. Wolfram Language & System Documentation Center. Retrieved from