Question

1. Define exp(iy) := cos(y) + i sin(y). a. Prove, using trigonometry, that exp(iy+iy') = exp(iy). expliy') for y, y' ER two real numbers. b. Prove directly (using Taylor series for sin and cos) that expliy) = " where n! denotes the factorial of n. Hint: you may use the fact that an infinite sum of complex numbers an converges if and only if Re(an) and Im(an) both converge and if it converges, an = Re(an) +i Im(an). Now apply this to an = (14)".

Hi, I really need help on both parts a and b of this Complex Analysis question. Thanks!

Answer 1

The given problem is to prove the following by using taylor's series..all the steps are clearly written in the pics...if you have any doubt ask in the comment section...THANK YOU :-)
bord ilyty'). 1) exp (iy) = cosytising ey-conytilny a) expiy tiy'). clytiy) cos(y+4') + i sin lyty) cosy, cosy' - sing king' til sin(y) Cosy' + cosy siny!) conycosylt i cosy kinyl - Sing Sing! tisory Cosy! - cosy (cosy't is my!) i Siny (Cosy tikiny) - (cosytising (Casiytising) a Cey) (e) = expliy). expliyi) Hence, exp (iytiyl) = expliy). explry) b) expliy) = 3 (g) expliy) - cosytising Tahtor series of cosy = 1- 4 + 4t - ys cosy pohy Taylor series of sny = x - y + 3 3 3 4 7. Sny & (-1 y 2nH no n! smy não (2nt)!

is we know that i ²-1 3 3 is it smyy- isung = iy - lys + i - ... (ig) + Liy)? - Cix)" + cup... (1) #cima ciga+kiye.. Losy tisíny = (0) +liy) *Cig; heter +Liye... - (1990 tlay)' + Lily couple we can write it as cosytising & n =0 n expliy) - & lei Hence forums