# Thread: Linear mappings of complex function

1. ## Linear mappings of complex function

This is from an introductory complex analysis textbook:

Express the given composition of mappings as a linear mapping $\displaystyle f(z)=az+b$.

(A) rotation through $\displaystyle \frac{\pi}{4}$, magnification by 2, and translation by $\displaystyle 1+i$​
(B) magnification by 2, translation by $\displaystyle \sqrt{2}$, and rotation through $\displaystyle \frac{\pi}{4}$
(C) translation by $\displaystyle \frac{\sqrt{2}}{2}$, rotation through $\displaystyle \frac{\pi}{4}$, then magnification by 2

The results are all supposed to turn out the same. I get that in (A) the answer is $\displaystyle 2e^{i\pi /4}+1+i$, but how do you do translations first and then rotation/magnification?

2. ## Re: Linear mappings of complex function

rotation: z = re^iθ → re^i(θ + a) = (e^ia)z , k = (e^ia)
magnification: z → kz, k real
translation: z → z + k, k complex

example: z → (z + k1) → k2(z + k1) → k3[k2(z + k1)]