TITLE: $n$th derivative of $e^{ax}\sin(bx+c)$
QUESTION [5 upvotes]: How can we substitute $r \cos \alpha$ and $r \sin \alpha$ for $a$ and $b$?
How, on successive differentiation, is there another $r$ multiplied?

REPLY [2 votes]: Hint
An easier solution is $$f(x)=e^{ax}\sin (bx+c)=\Im\{ e^{ax+jbx+jc}\}$$therefore $${d^n f(x)\over dx^n}=\Im\{{d^n\over dx^n} e^{(a+jb)x+jc}\}=\Im\{e^{jc}{d^n\over dx^n}e^{(a+jb)x}\}=\Im \{e^{jc}(a+jb)^ne^{(a+jb)x}\}$$