What is the Fourier Series Coefficient of F=exp(sin(x))?

syms x
T=2;
w0=2*pi/T;
f=exp(sin(x));
a0_sym=1/T*int(f,x,0,T);
a0_sym=double(a0_sym);

for n=1:3
    a_sym(n)=2/T*int(f*cos(n*w0*x),x,0,T);
    b_sym (n)=2/T*int(f*sin(n*w0*x),x,0,T);
    
    a(n)=double(a_sym(n));
    b(n)=double(b_sym(n)); 
end
s = a0_sym + sum(a.*cos((1:3).*w0.*x),2) + sum(b.*sin((1:3) .* w0.*x),2)

s = 

x_num = (-10:0.01:10).';
s_num = double(subs(s,x,x_num));
plot(x_num,s_num)

This is what I did but I'm not getting the desired result...

Please help

[Torsten]

And what is the "desired result" if it is not the result you get (see above) ?

[Bethel Eneh]

I am not very confident with the code i wrote above...

the expected result is

0. - 0.0221684i, -0.135748, 0. + 0.565159i, 1.26607, 0. -

0.565159i, -0.135748, 0. + 0.0221684i

[Torsten]

For a real-valued function, you'll never get complex Fourier series coefficients.

So the task seems to be different from what your code suggests.

Maybe you want to develop in exp(2*pi*1i*n*x/T) ?

[Bethel Eneh]

Alright thanks... The adjustment you made to the code helps. Thank you

[Bethel Eneh]

Please can you help me the code for fast fourier transform for F=exp(sin(x))?

[Torsten]

I have no experience with FFT, but

https://de.mathworks.com/help/matlab/ref/fft.html

should be the correct tool for this task.

Answer 1

[Bjorn Gustavsson]

Might I be so bold as to suggest (physicist writing about calculus here...) that this task ought to have some clever analytical solution method that is easier to do by hand than with help of computer. If you write the exponential as its series expansion:

For the first few terms we can easily rewrite (sin(t))^n into simple sin(n*t) and cos(n*t) terms, and there are trogonometric power-formulas: Trigonometric Power Formulas. If you use those you will get some gnarly expressions for the higher (or arbitrary) powers of (sin t)^k. If you plug those expression into matlab's symbolic tools you might get it to clean them up such that you can recognize the coefficients before the sin(k*2*pi*t) and cos(k*2*pi*t) terms. Those should be your Fourier-coefficients.

HTH