Formula:4532

Digital Library of Mathematical Functions ID 10.65.E3

${\displaystyle{\displaystyle\operatorname{ker}_{n}x=\tfrac{1}{2}(\tfrac{1}{2}x)^{-n}\sum_{k=0}^{n-1}\frac{(n-k-1)!}{k!}\cos\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k}-\ln\left(\tfrac{1}{2}x\right)\operatorname{ber}_{n}x+\tfrac{1}{4}\pi\operatorname{bei}_{n}x+\tfrac{1}{2}(\tfrac{1}{2}x)^{n}\sum_{k=0}^{\infty}\frac{\psi\left(k+1\right)+\psi\left(n+k+1\right)}{k!(n+k)!}\cos\left(\tfrac{3}{4}n\pi+\tfrac{1}{2}k\pi\right)(\tfrac{1}{4}x^{2})^{k},}}$

Symbols List

• ${\displaystyle{\displaystyle\operatorname{bei}_{\NVar{\nu}}\left(\NVar{x}\right)}}$: Kelvin function
• ${\displaystyle{\displaystyle\operatorname{ber}_{\NVar{\nu}}\left(\NVar{x}\right)}}$: Kelvin function
• ${\displaystyle{\displaystyle\operatorname{ker}_{\NVar{\nu}}\left(\NVar{x}\right)}}$: Kelvin function
• ${\displaystyle{\displaystyle\pi}}$: the ratio of the circumference of a circle to its diameter
• ${\displaystyle{\displaystyle\cos\NVar{z}}}$: cosine function
• ${\displaystyle{\displaystyle\psi\left(\NVar{z}\right)}}$: psi (or digamma) function
• ${\displaystyle{\displaystyle!}}$:
• ${\displaystyle{\displaystyle\ln\NVar{z}}}$: principal branch of logarithm function
• ${\displaystyle{\displaystyle n}}$: integer
• ${\displaystyle{\displaystyle k}}$: nonnegative integer
• ${\displaystyle{\displaystyle x}}$: real variable
 

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