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:

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:

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 :

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:

First derivative:

Higher derivatives:

Plot higher derivatives for order :

Formula for the derivative:

Indefinite integral of BesselK:

Integrate expressions involving BesselK:

Definite integral of BesselK over its real domain:

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:

LaplaceTransform:

HankelTransform:

Compute the Mellin transform using MellinTransform:

Use FullSimplify to simplify Bessel functions:

Verify the identity :

Recurrence relations :

Integral representation of BesselK:

Represent using BesselI and Sin:

BesselK can be represented in terms of MeijerG:

BesselK can be represented as a DifferenceRoot: