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Question Number 25822 by abdo imad last updated on 15/Dec/17

a n s w e r t o t h e q u e s t i o n o f p (X) = {(1 + i X)}^{n} - {(1 - i X)}^{n}

k e y o f s o l u t i o n a f t e r r e s o l v i n g p (X) = 0 t h e r o o t s o f p (X)

a r e x_{k} = t a n (k π / n) w i t h k i n [[0 . n - 1]] s o

p (X) = \propto \prod_{k = 0}^{k = n - 1} (X - x_{k^{}}) l e t s e a r s h \propto b y u s i n g b i n o m e f o r m u l a

p (X) = 2 i \sum_{p = 0}^{} {(- 1)}^{p} C_{n}^{2 p + 1} X^{2 p + 1} s o

\propto= 2 i {(- 1)}^{[n - 1 / 2]} C_{n}^{}

c a s e 1 n = 2 N p (X) =\propto \prod_{k = 0}^{k = 2 N - 1} (X - t a n (k π / 2 N))

a n d \propto= 4 i n {(- 1)}^{N - 1}

c a s e 2 n = 2 N + 1 p (X) =\propto \prod_{k = 0}^{k = 2 N} (X - t a n (k π / 2 N + 1)

a n d \propto= 2 i {(- 1)}^{N}