# Approximate $h$ in $F(\theta)=\sin \theta \int_{-L}^{+L}h(z)e^{-ikz\cos \theta} \,dz$

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.

• Is $F(\theta)$ real? Jan 5, 2015 at 15:10
• @user7257 We don't know, but what will happen if you consider it real? Jan 5, 2015 at 15:12
• How would the problem look like for complex $z$? Will the integral still be along a simple line?
– Dirk
Jan 5, 2015 at 16:26

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.