Difference between revisions of "Relationship between Airy Bi and modified Bessel I"
From specialfunctionswiki
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| − | + | ==Theorem== | |
The following formula holds: | The following formula holds: | ||
$$\mathrm{Bi}(z)=\sqrt{\dfrac{z}{3}} \left( I_{\frac{1}{3}}\left(\frac{2}{3}x^{\frac{3}{2}} \right) + I_{-\frac{1}{3}} \left( \frac{2}{3} x^{\frac{3}{2}} \right) \right),$$ | $$\mathrm{Bi}(z)=\sqrt{\dfrac{z}{3}} \left( I_{\frac{1}{3}}\left(\frac{2}{3}x^{\frac{3}{2}} \right) + I_{-\frac{1}{3}} \left( \frac{2}{3} x^{\frac{3}{2}} \right) \right),$$ | ||
where $\mathrm{Bi}$ denotes the [[Airy Bi]] function and $I_{\nu}$ denotes the [[Modified Bessel I sub nu|modified Bessel $I$]]. | where $\mathrm{Bi}$ denotes the [[Airy Bi]] function and $I_{\nu}$ denotes the [[Modified Bessel I sub nu|modified Bessel $I$]]. | ||
| − | === | + | ==Proof== |
| + | |||
| + | ==References== | ||
[[Category:Theorem]] | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 07:16, 16 June 2016
Theorem
The following formula holds: $$\mathrm{Bi}(z)=\sqrt{\dfrac{z}{3}} \left( I_{\frac{1}{3}}\left(\frac{2}{3}x^{\frac{3}{2}} \right) + I_{-\frac{1}{3}} \left( \frac{2}{3} x^{\frac{3}{2}} \right) \right),$$ where $\mathrm{Bi}$ denotes the Airy Bi function and $I_{\nu}$ denotes the modified Bessel $I$.
