Function Repository Resource:

# BickleyKi

Evaluate the Bickley function

Contributed by: Jan Mangaldan
 ResourceFunction["BickleyKi"][n,z] gives the Bickley function Kin(z).

## Details and Options

Mathematical function, suitable for both symbolic and numerical manipulation.
The Bickley function satisfies , where K0(t) is the modified Bessel function BesselK.
ResourceFunction["BickleyKi"][n,z] has a branch cut discontinuity in the complex z plane running from - to 0.
For certain special arguments, ResourceFunction["BickleyKi"] automatically evaluates to exact values.
ResourceFunction["BickleyKi"] can be evaluated to arbitrary numerical precision.

## Examples

### Basic Examples (3)

Evaluate numerically:

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Plot Ki1(z):

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Series at the origin:

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### Scope (4)

Evaluate for complex arguments and orders:

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Evaluate to high precision:

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

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### Applications (4)

Plot a complex-ordered Bickley function over the complex plane:

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Average probability that a neutron travels across two parallel lines separated by a distance h without a collision:

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Compare with the integral representation:

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Express a modified Bessel function of the second kind as a finite sum of Bickley functions:

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### Properties and Relations (5)

For n=0, BickleyKi is equal to K0(z):

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For n>0, BickleyKi is equal to an iterated integral of K0(z):

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For n<0, BickleyKi is equal to (-1)nd-nK0(z)/dz-n:

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Express a Bickley function of noninteger order in terms of simpler functions:

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Compare BickleyKi with the integral definition:

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### Possible Issues (2)

Machine precision is not sufficient to obtain the correct result:

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## Version History

• 2.0.0 – 11 March 2020
• 1.0.0 – 03 December 2019