# modified Bessel equation

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##### 3: 10.72 Mathematical Applications
###### §10.72(i) Differential Equations with Turning Points
Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter. …
##### 4: 10.73 Physical Applications
10.73.3 $\nabla^{4}W+\lambda^{2}\frac{{\partial}^{2}W}{{\partial t}^{2}}=0.$
##### 5: 10.34 Analytic Continuation
10.34.1 $I_{\nu}\left(ze^{m\pi i}\right)=e^{m\nu\pi i}I_{\nu}\left(z\right),$
10.34.2 $K_{\nu}\left(ze^{m\pi i}\right)=e^{-m\nu\pi i}K_{\nu}\left(z\right)-\pi i\sin% \left(m\nu\pi\right)\csc\left(\nu\pi\right)I_{\nu}\left(z\right).$
10.34.4 $K_{\nu}\left(ze^{m\pi i}\right)=\csc\left(\nu\pi\right)\left(\pm\sin\left(m\nu% \pi\right)K_{\nu}\left(ze^{\pm\pi i}\right)\mp\sin\left((m\mp 1)\nu\pi\right)K% _{\nu}\left(z\right)\right).$
10.34.5 $K_{n}\left(ze^{m\pi i}\right)=(-1)^{mn}K_{n}\left(z\right)+(-1)^{n(m-1)-1}m\pi iI% _{n}\left(z\right),$
10.34.6 $K_{n}\left(ze^{m\pi i}\right)=\pm(-1)^{n(m-1)}mK_{n}\left(ze^{\pm\pi i}\right)% \mp(-1)^{nm}(m\mp 1)K_{n}\left(z\right).$
##### 6: 10.39 Relations to Other Functions
10.39.3 $K_{\frac{1}{4}}\left(z\right)=\pi^{\frac{1}{2}}z^{-\frac{1}{4}}U\left(0,2z^{% \frac{1}{2}}\right),$
10.39.5 $I_{\nu}\left(z\right)=\frac{(\tfrac{1}{2}z)^{\nu}e^{\pm z}}{\Gamma\left(\nu+1% \right)}M\left(\nu+\tfrac{1}{2},2\nu+1,\mp 2z\right),$
10.39.6 $K_{\nu}\left(z\right)=\pi^{\frac{1}{2}}(2z)^{\nu}e^{-z}U\left(\nu+\tfrac{1}{2}% ,2\nu+1,2z\right),$
##### 7: 10.30 Limiting Forms
10.30.4 $I_{\nu}\left(z\right)\sim e^{z}/\sqrt{2\pi z},$ $|\operatorname{ph}z|\leq\tfrac{1}{2}\pi-\delta$,
10.30.5 $I_{\nu}\left(z\right)\sim e^{\pm(\nu+\frac{1}{2})\pi i}e^{-z}/\sqrt{2\pi z},$ $\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{3}{2}\pi-\delta$.
##### 8: 10.31 Power Series
10.31.1 $K_{n}\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)^{-n}\sum_{k=0}^{n-1}\frac{(n-k% -1)!}{k!}(-\tfrac{1}{4}z^{2})^{k}+(-1)^{n+1}\ln\left(\tfrac{1}{2}z\right)I_{n}% \left(z\right)+(-1)^{n}\tfrac{1}{2}(\tfrac{1}{2}z)^{n}\sum_{k=0}^{\infty}\left% (\psi\left(k+1\right)+\psi\left(n+k+1\right)\right)\frac{(\tfrac{1}{4}z^{2})^{% k}}{k!(n+k)!},$
10.31.2 $K_{0}\left(z\right)=-\left(\ln\left(\tfrac{1}{2}z\right)+\gamma\right)I_{0}% \left(z\right)+\frac{\tfrac{1}{4}z^{2}}{(1!)^{2}}+(1+\tfrac{1}{2})\frac{(% \tfrac{1}{4}z^{2})^{2}}{(2!)^{2}}+(1+\tfrac{1}{2}+\tfrac{1}{3})\frac{(\tfrac{1% }{4}z^{2})^{3}}{(3!)^{2}}+\dotsi.$
##### 9: 10.32 Integral Representations
10.32.1 $I_{0}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{\pm z\cos\theta}\,\mathrm{d}% \theta=\frac{1}{\pi}\int_{0}^{\pi}\cosh\left(z\cos\theta\right)\,\mathrm{d}\theta.$
10.32.3 $I_{n}\left(z\right)=\frac{1}{\pi}\int_{0}^{\pi}e^{z\cos\theta}\cos\left(n% \theta\right)\,\mathrm{d}\theta.$
10.32.6 $K_{0}\left(x\right)=\int_{0}^{\infty}\cos\left(x\sinh t\right)\,\mathrm{d}t=% \int_{0}^{\infty}\frac{\cos\left(xt\right)}{\sqrt{t^{2}+1}}\,\mathrm{d}t,$ $x>0$.
10.32.9 $K_{\nu}\left(z\right)=\int_{0}^{\infty}e^{-z\cosh t}\cosh\left(\nu t\right)\,% \mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$.
10.32.10 $K_{\nu}\left(z\right)=\tfrac{1}{2}(\tfrac{1}{2}z)^{\nu}\int_{0}^{\infty}\exp% \left(-t-\frac{z^{2}}{4t}\right)\frac{\,\mathrm{d}t}{t^{\nu+1}},$ $|\operatorname{ph}z|<\tfrac{1}{4}\pi$.
##### 10: 10.45 Functions of Imaginary Order
With $z=x$, and $\nu$ replaced by $i\nu$, the modified Bessel’s equation (10.25.1) becomes …
10.45.3 $\displaystyle\widetilde{I}_{-\nu}\left(x\right)=\widetilde{I}_{\nu}\left(x% \right),$ $\displaystyle\widetilde{K}_{-\nu}\left(x\right)=\widetilde{K}_{\nu}\left(x% \right),$
10.45.6 $\widetilde{I}_{\nu}\left(x\right)=\left(\frac{\sinh\left(\pi\nu\right)}{\pi\nu% }\right)^{\frac{1}{2}}\cos\left(\nu\ln\left(\tfrac{1}{2}x\right)-\gamma_{\nu}% \right)+O\left(x^{2}\right),$
10.45.7 $\widetilde{K}_{\nu}\left(x\right)=-\left(\frac{\pi}{\nu\sinh\left(\pi\nu\right% )}\right)^{\frac{1}{2}}\*\sin\left(\nu\ln\left(\tfrac{1}{2}x\right)-\gamma_{% \nu}\right)+O\left(x^{2}\right),$
10.45.8 $\widetilde{K}_{0}\left(x\right)=K_{0}\left(x\right)=-\ln\left(\tfrac{1}{2}x% \right)-\gamma+O\left(x^{2}\ln x\right),$