I am a little stuck at the moment with the Fourier transform of a function $w(z)$ that is defined piecewise and discontinuous at $z = 0$: $w\left(z\right) = \begin{cases} -\gamma_1 e^{-\gamma_{1}z} & z > 0 \\ \gamma_2 e^{\gamma_{2}z} & z < 0\end{cases}$ with $\gamma_i \geq 0$.
In a naive attempt, I calculated the following: $ W \left( k \right) = \int_{-\infty}^{\infty}w(z) e^{-\mathrm{i}k z}dz$
$ = -\gamma_{1}\int_{0}^{\infty}e^{-\left(\gamma_{1}+\mathrm{i}k\right)z}dz+\gamma_{2}\int_{-\infty}^{0}e^{\left(\gamma_{2}-\mathrm{i}k\right)z}dz = -\frac{\gamma_{1}}{\left(\gamma_{1}+\mathrm{i}k\right)}+\frac{\gamma_{2}}{\left(\gamma_{2}-\mathrm{i}k\right)}$
But I am almost sure that I miscalculated since there should somewhere occur a $\delta$-contribution proportional to $\left(\gamma_2 - \gamma_1 \right)\cdot\delta(0)$ as in the case of the Heaviside function.
Hence:
Where is my brain-overflow in the calculation given?
Thank you in advance
Sincerely
Robert
PS.: Even though it is not such a thing, I marked this question as homework since it is on this level.