I want to find the image of the function that defined as $\exp(x+iy)=e^x(\cos(y)+i\sin(y))$ for $z\in L$ where $L=\left\{z\in \mathbb C \mid a\mathrm{Re}(z)+b\mathrm{Im}(z)+c=0 \right\}$ and $a^2+b^2\ne 0$.
I tried to get to an expression for $z$ uing the identities $\mathrm{Re}(z)=\dfrac{z+\bar z}{2}$ and $\mathrm{Im}(z)=\dfrac{z-\bar z}{2i}$ but I got to a dead end.
How should I approach this problem?
If $b\neq0$, then $ax+by+c=0\iff y=-\frac bax-\frac ca$ and therefore\begin{align}\exp(x+yi)&=\exp\left(x-\left(\frac bax+\frac ca\right)i\right)\\&=e^x\left(\cos\left(\frac bax+\frac ca\right)-\sin\left(\frac bax+\frac ca\right)\right).\end{align}This is a spiral. Can you deal with the case $b=0$?