Fourier series using Bessel function












2












$begingroup$


so Im stuck on the following problem;



Use the identity



$exp(ixsintheta) = sumlimits_{k=-infty}^infty J_k(x)exp(iktheta)$



to find the Fourier series of $cos(theta + 4sintheta)$, where $J_k$ is the $k$th Bessel functions.



I tried to expand $cos(theta + 4sintheta)$ using trig identities and use it to find $A_n$ and $B_n$ the usual way for Fourier series, but this lead to an ugly integral that maple cant even solve. Where does the identity fit in? Even anyone has any hints to start it would be greatly appreciated!



Update: continue using Mhenni's hint



Try to compute $A_n$:



$cos(theta+4sin t) = (1/2)(exp(i(t+4sin t))+exp(-i(t+4sin t) $



$= (1/2)(e^{it}sum J_k(4)e^{ikt} + e^{-it}sum J_k(-4)e^{ikt})$



$A_n = (1/2pi)intlimits_{-pi}^{pi}sum(J_k(4)e^{it+ikt}+J_k(-4)e^{-it+ikt})cos(mt)dt$



is this looking any better? or have I gone wrong anywhere? as when I integrate this using maple it looks pretty bad too.






fourier-series bessel-functions







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Use the identity $cos t=(e^{it} + e^{-it}) /2$.
    $endgroup$
    – Mhenni Benghorbal
    Jan 28 '15 at 23:57


















2












$begingroup$


so Im stuck on the following problem;



Use the identity



$exp(ixsintheta) = sumlimits_{k=-infty}^infty J_k(x)exp(iktheta)$



to find the Fourier series of $cos(theta + 4sintheta)$, where $J_k$ is the $k$th Bessel functions.



I tried to expand $cos(theta + 4sintheta)$ using trig identities and use it to find $A_n$ and $B_n$ the usual way for Fourier series, but this lead to an ugly integral that maple cant even solve. Where does the identity fit in? Even anyone has any hints to start it would be greatly appreciated!



Update: continue using Mhenni's hint



Try to compute $A_n$:



$cos(theta+4sin t) = (1/2)(exp(i(t+4sin t))+exp(-i(t+4sin t) $



$= (1/2)(e^{it}sum J_k(4)e^{ikt} + e^{-it}sum J_k(-4)e^{ikt})$



$A_n = (1/2pi)intlimits_{-pi}^{pi}sum(J_k(4)e^{it+ikt}+J_k(-4)e^{-it+ikt})cos(mt)dt$



is this looking any better? or have I gone wrong anywhere? as when I integrate this using maple it looks pretty bad too.






fourier-series bessel-functions







share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Use the identity $cos t=(e^{it} + e^{-it}) /2$.
    $endgroup$
    – Mhenni Benghorbal
    Jan 28 '15 at 23:57
















2












2








2





$begingroup$


so Im stuck on the following problem;



Use the identity



$exp(ixsintheta) = sumlimits_{k=-infty}^infty J_k(x)exp(iktheta)$



to find the Fourier series of $cos(theta + 4sintheta)$, where $J_k$ is the $k$th Bessel functions.



I tried to expand $cos(theta + 4sintheta)$ using trig identities and use it to find $A_n$ and $B_n$ the usual way for Fourier series, but this lead to an ugly integral that maple cant even solve. Where does the identity fit in? Even anyone has any hints to start it would be greatly appreciated!



Update: continue using Mhenni's hint



Try to compute $A_n$:



$cos(theta+4sin t) = (1/2)(exp(i(t+4sin t))+exp(-i(t+4sin t) $



$= (1/2)(e^{it}sum J_k(4)e^{ikt} + e^{-it}sum J_k(-4)e^{ikt})$



$A_n = (1/2pi)intlimits_{-pi}^{pi}sum(J_k(4)e^{it+ikt}+J_k(-4)e^{-it+ikt})cos(mt)dt$



is this looking any better? or have I gone wrong anywhere? as when I integrate this using maple it looks pretty bad too.






fourier-series bessel-functions







share|cite|improve this question











$endgroup$




so Im stuck on the following problem;



Use the identity



$exp(ixsintheta) = sumlimits_{k=-infty}^infty J_k(x)exp(iktheta)$



to find the Fourier series of $cos(theta + 4sintheta)$, where $J_k$ is the $k$th Bessel functions.



I tried to expand $cos(theta + 4sintheta)$ using trig identities and use it to find $A_n$ and $B_n$ the usual way for Fourier series, but this lead to an ugly integral that maple cant even solve. Where does the identity fit in? Even anyone has any hints to start it would be greatly appreciated!



Update: continue using Mhenni's hint



Try to compute $A_n$:



$cos(theta+4sin t) = (1/2)(exp(i(t+4sin t))+exp(-i(t+4sin t) $



$= (1/2)(e^{it}sum J_k(4)e^{ikt} + e^{-it}sum J_k(-4)e^{ikt})$



$A_n = (1/2pi)intlimits_{-pi}^{pi}sum(J_k(4)e^{it+ikt}+J_k(-4)e^{-it+ikt})cos(mt)dt$



is this looking any better? or have I gone wrong anywhere? as when I integrate this using maple it looks pretty bad too.






fourier-series bessel-functions




fourier-series bessel-functions






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 29 '15 at 2:39







Mr Man

















asked Jan 28 '15 at 23:45









Mr ManMr Man

404




404








  • 1




    $begingroup$
    Use the identity $cos t=(e^{it} + e^{-it}) /2$.
    $endgroup$
    – Mhenni Benghorbal
    Jan 28 '15 at 23:57
















  • 1




    $begingroup$
    Use the identity $cos t=(e^{it} + e^{-it}) /2$.
    $endgroup$
    – Mhenni Benghorbal
    Jan 28 '15 at 23:57










1




1




$begingroup$
Use the identity $cos t=(e^{it} + e^{-it}) /2$.
$endgroup$
– Mhenni Benghorbal
Jan 28 '15 at 23:57






$begingroup$
Use the identity $cos t=(e^{it} + e^{-it}) /2$.
$endgroup$
– Mhenni Benghorbal
Jan 28 '15 at 23:57












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