Modified Bessel K

The modified Bessel function of the second kind is defined by $$K_{\nu}(z)=\dfrac{\pi}{2} \dfrac{I_{-\nu}(z)-I_{\nu}(z)}{\sin(\nu \pi)},$$ where $I_{\nu}$ is the modified Bessel function of the first kind.

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  Domain coloring of $K_1$.

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  Modified Bessel functions from Abramowitz&Stegun.

Properties

Proposition: The following formula holds: $$K_{\frac{1}{2}}(z)=\sqrt{\dfrac{\pi}{2}}\dfrac{e^{-z}}{\sqrt{z}}; z>0.$$

Proof: █

Theorem

The following formula holds: $$\mathrm{Ai}(z)=\dfrac{1}{\pi} \sqrt{\dfrac{z}{3}} \mathrm{K}_{\frac{1}{3}} \left( \dfrac{2}{3} x^{\frac{3}{2}} \right),$$ where $\mathrm{Ai}$ is the Airy Ai function and $K_{\nu}$ denotes the modified Bessel $K$.

References

<center>Bessel functions

Anger function

Bessel-Clifford

Bessel $J_{\nu}$

Bessel $Y_{\nu}$

Bessel $I_{\nu}$

Modified Bessel $K_{\nu}$

Spherical Bessel $j_{\nu}$

Spherical Bessel $y_{\nu}$

</center>

References

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