# Determing an inverse Fourier transform

The inverse Fourier transform is defined as:

$$\mathcal{F}^{-1}[g](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} g(k) e^{i k x} d k$$

I can't get an inverse Fourier Transform to

Q1: $$g(k)=\cos (\sqrt{ k^2 d -a^2 })$$ Q2:
$$g(k)=\frac{1}{ \sqrt{ k^2 d -a^2 }}\sin (\sqrt{ k^2 d -a^2 })$$

I would really appreciate some any help .

• why would you need a closed form expression for the Fourier transform of those complicated function ? – reuns Apr 17 '16 at 19:38
• have you tried mathematica or stuff like that? – user190080 Apr 17 '16 at 19:42
• No. Is Mathematica obtain the result ? – Hamada Al Apr 20 '16 at 10:11

Q1: $$\mathcal{F}^{-1}[g](x) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{\infty} \cos (\sqrt{ k^2 d -a^2 }) e^{i k x} d k$$ I see no reason to think it may be written any simpler than that.
• Do you mean that i should write $2 \cos(\theta)= Re \quad( e^{i \theta})$ or what ? – Hamada Al Apr 17 '16 at 23:23
• For example when $k=0$ you are talking about $\cos(\sqrt{-a^2})$. – GEdgar Apr 18 '16 at 0:55