find-n-0-sin-na-sina-n-x-n-n-with-0-lt-a-lt-pi-

Question Number 33127 by prof Abdo imad last updated on 10/Apr/18

f i n d \sum_{n = 0}^{\infty} \frac{s i n (n a)}{{(s i n a)}^{n}} \frac{x^{n}}{n!} w i t h 0 < a < π .

Commented by prof Abdo imad last updated on 12/Apr/18

S (x) = I m (\sum_{n = 0}^{\infty} \frac{e^{i n a} x^{n}}{n! {(s i n a)}^{n}}) b u t

w (x) = \sum_{n = 0}^{\infty} \frac{e^{i n a} x^{n}}{n! {(s i n a)}^{n}} = \sum_{n = 0}^{\infty} \frac{1}{n!} {(\frac{e^{i a} x}{s i n a})}^{n}

b u t w e h a v e \sum_{n = 0}^{\infty} \frac{u^{n}}{n!} = e^{u} \forall u \in C \Rightarrow

w (x) = e^{\frac{e^{i a}}{s i n a} x} = e^{\frac{x}{s i n a} (c o a + i s i n a)}

= e^{\frac{x c o s a}{s i n s}} e^{i x} = e^{\frac{x c o s a}{s i n a}} (c o s x + i s i n x) \Rightarrow

S (x) = s i n x e^{\frac{x c o s a}{s i n a}} .