Negative Gamma?

Calculus Level 4

$\large {\Gamma {(-1.5)}=?}$

Enter $1.729$ , if you think that Gamma function is not defined for $-1.5$ .

1 solution

Nihar Mahajan
Mar 3, 2016

We use the property $\Gamma(n+1)=n\Gamma(n)$ two times. Hence we have:

$\Gamma\left(\dfrac{-1}{2}\right)=-\dfrac{3}{2}\Gamma\left(\dfrac{-3}{2}\right) \dots \quad (1)$

$\Gamma\left(\dfrac{1}{2}\right)=-\dfrac{1}{2}\Gamma\left(\dfrac{-1}{2}\right) \dots \quad (2)$

Substituting $(1)$ in $(2)$ ,

$\Gamma\left(\dfrac{1}{2}\right)=-\dfrac{1}{2} \times -\dfrac{3}{2}\Gamma\left(\dfrac{-3}{2}\right)$

$\Rightarrow \sqrt{\pi} = \dfrac{3}{4} \Gamma(-1.5)$

$\Rightarrow \Gamma(-1.5) = \dfrac{4\sqrt{\pi}}{3} \approx \boxed{2.36}$

Good explanation.

I used Euler reflection formula!

Aniket Sanghi - 4 years, 8 months ago

The gamma function is defined for all complex numbers except the non-positive integers. So this one needed to be reported

Chirag Shyamsundar - 4 years, 7 months ago

Same approach

Aakash Khandelwal - 5 years, 3 months ago

Same way dude !!!

abc xyz - 5 years, 3 months ago