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Consider $$F(\theta)=\sin \theta \int_{-L}^{+L}h(z)e^{-ikz\cos \theta} \,dz$$ $$|z|\le L$$ $$0 \le \theta \le \pi$$ By having knowledge of $F(\theta)$, how can one approximate $h(z)$? In addition, I know that $F$ is differentiable with respect to $\theta$.

How can I model and solve such problem with unknown function $h(z)$ in Mathematica or any other programming language?

There is an idea in which we write Fourier series for both $F$ and h then by solving or manipulation of the integral we approximate the coefficients.

** $k$ is the wave number, don't confuse it with anything else.
** $z$ might be a complex number, but in this case you can consider it as real.
**Any idea on this problem would be extremely appreciated.

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  • $\begingroup$ Is $F(\theta)$ real? $\endgroup$ – user7257 Jan 5 '15 at 15:10
  • $\begingroup$ @user7257 We don't know, but what will happen if you consider it real? $\endgroup$ – FreeMind Jan 5 '15 at 15:12
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    $\begingroup$ How would the problem look like for complex $z$? Will the integral still be along a simple line? $\endgroup$ – Dirk Jan 5 '15 at 16:26
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This is a linear integral equation (because the right hand side is linear in $h$) and presumably the problem is ill-posed (for one, because of the multiplication by $\sin(\theta)$ but also because the "integral kernel" is smooth).

Methods to solve this (approximately) are general Galerkin methods or collocation methods. For Galerkin methods you take a finite linear independent set (i.e. a basis for a finite dimensional subspace) of functions to model $h$ and $F$ and derive the linear system for the coefficients of $h$ and $F$ for this basis. For collocation methods you discretize the integral (e.g. by trapezoidal rule) and write down the (linear) equations that you get if you evaluate $F$ at some points.

By warned that the respective linear systems that you will get may be ill-conditioned and that you should use some regularization method (check any book on linear inverse problems or look up "Tikhonov regularization", "truncated SVD", "Landweber iteration" or "regularization by CG)."

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  • $\begingroup$ Thanks for your response, I wish you could perform some calculations, I have found that Filon's method is another way to tackle it, but the hard thing is to apply the method to the problem. $\endgroup$ – FreeMind Jan 6 '15 at 18:19
  • $\begingroup$ You may discretize the integral basicall as you like. I suspect lots of relevant unmentioned details to lurk behind the scene so I guess it is not a good idea to have somebody else to do some test computations (not that I had the time, anyway…). $\endgroup$ – Dirk Jan 8 '15 at 13:37
  • $\begingroup$ Could you please show me some calculation? How do you write the integral in discrete form?! $\endgroup$ – FreeMind Jan 8 '15 at 23:44

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