Difference between revisions of "Relationship between Airy Ai and modified Bessel K"
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| − | + | ===Theorem=== | |
| − | + | The following formula holds: | |
$$\mathrm{Ai}(z)=\dfrac{1}{\pi} \sqrt{\dfrac{z}{3}} \mathrm{K}_{\frac{1}{3}} \left( \dfrac{2}{3} x^{\frac{3}{2}} \right),$$ | $$\mathrm{Ai}(z)=\dfrac{1}{\pi} \sqrt{\dfrac{z}{3}} \mathrm{K}_{\frac{1}{3}} \left( \dfrac{2}{3} x^{\frac{3}{2}} \right),$$ | ||
where $\mathrm{Ai}$ is the [[Airy Ai]] function and $K_{\nu}$ denotes the [[Modified Bessel K sub nu|modified Bessel $K$]]. | where $\mathrm{Ai}$ is the [[Airy Ai]] function and $K_{\nu}$ denotes the [[Modified Bessel K sub nu|modified Bessel $K$]]. | ||
| − | + | ===Proof=== | |
| − | + | ||
| − | + | [[Category:Theorem]] | |
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Revision as of 08:02, 5 June 2016
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 $K$.
