Nice-Calculus-prove-that-x-n-1-a-n-sin-nx-n-e-acos-x-sin-asin-x-m-n-

Question Number 141057 by mnjuly1970 last updated on 15/May/21

\dots . . N i c e \dots \dots \dots \dots C a l c u l u s \dots . .

p r o v e t h a t :

Ω (x) := \underset{n = 1}{\sum^{\infty}} a^{n} . \frac{s i n (n x)}{n!} = e^{a c o s (x)} s i n (a s i n (x))

\dots . m . n

Answered by Dwaipayan Shikari last updated on 15/May/21

\frac{1}{2 i} \underset{n = 1}{\sum^{\infty}} a^{n} \frac{e^{i n x}}{n!} - a^{n} \frac{e^{- i n x}}{n!}

= \frac{1}{2 i} (e^{{a e}^{i x}} - e^{{a e}^{- i x}}) = \frac{1}{2 i} (e^{a c o s (x) + a i s i n (x)} - e^{a c o s (x) - a i s i n (x)})

= \frac{e^{a c o s (x)}}{2 i} (e^{a i s i n (x)} - e^{- a i s i n (x)}) = s i n (a s i n x) e^{a c o s x}

Commented by mnjuly1970 last updated on 15/May/21

t h a n k s a l o t . .

Commented by Dwaipayan Shikari last updated on 15/May/21

: -)

Answered by mindispower last updated on 15/May/21

Ω (x) = I m \sum_{n ⩾ 1} \frac{a^{n} e^{i n x}}{n!} = I m \sum_{n ⩾ 1} \frac{{({a e}^{i x})}^{n}}{n!}, i f a \in R

= I m \sum_{n ⩾ 0} \frac{{({a e}^{i x})}^{n}}{n!} = I m (e^{{a e}^{i x}})

= I m (e^{a c o s (x)} (c o s (a x) + i s i n (a x))

= e^{a c o s (x)} s i n (s i n (a x))

Commented by mnjuly1970 last updated on 15/May/21

g r a t e f u l . .

Commented by mindispower last updated on 15/May/21

p l e a s u r