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Question Number 34698 by abdo imad last updated on 10/May/18

l e t f (x) = e^{x} s i n x . d e v e l o p p f a t i n t e g r s e r i e .

Commented by abdo mathsup 649 cc last updated on 10/May/18

f (x) = I m (e^{x} e^{i x}) = I m (e^{(1 + i) x}) b u t

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

= \sum_{n = 0}^{\infty} \frac{{\sqrt{2} e^{i \frac{π}{4}}}^{n}}{n!} x^{n}

= \sum_{n = 0}^{\infty} \frac{{(\sqrt{2})}^{n} e^{i n \frac{π}{4}}}{n!} x^{n} \Rightarrow

f (x) = \sum_{n = 0}^{\infty} \frac{{(\sqrt{2})}^{n}}{n!} s i n (n \frac{π}{4}) x^{n} .

Commented by abdo mathsup 649 cc last updated on 10/May/18

a n o t h e r m e t h o d

w e h a v e ? f (x) = \sum_{n = 0}^{\infty} \frac{f^{(n)} (0)}{n!} x^{n} b u t l e i b n i z

f o r m u l a g i v e f^{(n)} (x) = \sum_{k = 0}^{n} C_{n}^{k} {(s i n x)}^{(k)} {(e^{x})}^{(n - k)}

= \sum_{k = 0}^{n} C_{n}^{k} s i n (x + k \frac{π}{2}) e^{x} \Rightarrow

f^{(n)} (0) = \sum_{k = 0}^{n} C_{n}^{k} s i n (k \frac{π}{2}) \Rightarrow

f (x) = \sum_{n = 0}^{\infty} \frac{1}{n!} (\sum_{k = 0}^{n} C_{n}^{k} s i n (k \frac{π}{2})) x^{n} .

Answered by tanmay.chaudhury50@gmail.com last updated on 10/May/18

p = e^{x} c o s x q = e^{x} s i n x

p + i q = e^{x} c o s x + {i e}^{x} s i n x

= e^{x} . e^{i x}

= e^{x (1 + i)}

= x (1 + i) + \frac{x^{2} . {(1 + i)}^{2}}{2!} + \frac{x^{3} {(1 + i)}^{3}}{3!} + \frac{x^{4} {(1 + i)}^{4}}{4!} + \dots

= (x + i x) + \frac{(i 2 x^{2})}{2!} + x^{3} (1 + 3 i - 3 - i) / 3! + x^{4} \frac{(- 4)}{4 -!} + . .

s o e^{x} s i n x =

x + \frac{2 x^{2}}{2!} + x^{3} \frac{2}{3!} + \dots .