complex-analysis-if-f-n-x-d-n-dx-n-x-x-C-C-0-n-C-Z-0-and-g-n-x-0-1-f-n-x-d-then-find-the-value-of-

Question Number 137275 by mnjuly1970 last updated on 31/Mar/21

\dots \dots c o m p l e x a n a l y s i s \dots . .

i f, f (α, n, x) = \frac{d^{n}}{{d x}^{n}} (α^{x}), x \in C

α \in C - {0}, n \in C - Z^{-} \cup {0}

a n d g (n, x) = \int_{0}^{1} f (α, n, x) d α

t h e n f i n d t h e v a l u e o f \dots

Ω = \frac{I m (g (i, 0))}{R e (Γ (i))}

s o l u t i o n :

g (n, x) = \int_{0}^{1} \frac{d^{n}}{{d x}^{n}} (α^{x}) d α

= \int_{0}^{1} \frac{d^{n}}{{d x}^{n}} (e^{x l n (α)}) d α \dots . . ⟨ * ⟩

\frac{d^{n}}{{d x}^{n}} (e^{x l n (α)}) = {(l n (α))}^{n} α^{x}

⟨ * ⟩ \to \dots g (n, x) = \int_{0}^{1} {(l n (α))}^{n} α^{x} d α

\underset{α = e^{- y}}{\overset{l n (α) = - y}{=}} \int_{0}^{\infty} {(- 1)}^{n} y^{n} e^{- y x} e^{- y} d y

= {(- 1)}^{n} \int_{0}^{1} y^{n} . e^{- y (1 + x)} d y

\overset{y (1 + x) = t}{=} {(- 1)}^{n} \int_{0}^{1} \frac{t^{n} e^{- t}}{{(1 + x)}^{n + 1}} d t

= e^{i n π} . \frac{1}{{(1 + x)}^{n + 1}} . Γ (n + 1)

g (i, 0) = e^{- π} . i . Γ (i) \dots \dots ⟨ * * ⟩

Γ (i) \in C \Rightarrow Γ (i) = R e (Γ (i)) + I m (Γ (i))

⟨ * * ⟩ \to \dots g (i, 0) = e^{- π} . i . [R e (Γ (i)) + I m (Γ (i))]

∴ Ω = \frac{e^{- π} R e (Γ (i))}{R e (Γ (i))} = e^{- π} \dots ✓ ✓