A calculus problem by Kaleem Kħặŋ

Calculus Level 2

( ( 0.5 ) ! × 2 ) 2 = ? \large \left((0.5)! \times 2 \right)^2 = \, ?

√2 π i e

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1 solution

Oon Han
Jan 5, 2019

Using the gamma function: ( 0.5 ) ! = Γ ( 3 2 ) = 0 x e x d x = π 2 (0.5)!=\Gamma \left( \frac { 3 }{ 2 } \right) =\int _{ 0 }^{ \infty }{ \frac { \sqrt { x } }{ { e }^{ x } } } dx=\frac { \sqrt { \pi } }{ 2 } I won't go into the details of this calculation but here's the calculation of the integral from a calculator. Anyway, from here, ( ( 0.5 ) ! × 2 ) 2 = ( π 2 × 2 ) 2 = π { \left((0.5)!\times 2\right) }^{ 2 }={ \left( \frac { \sqrt { \pi } }{ 2 } \times 2 \right) }^{ 2 }=\pi and we're done.

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