$\Gamma(\text{problem})$

Calculus Level 3

We know from gamma function that $\begin{aligned} \Gamma(2) &= 1 \\ \Gamma(3) &= 2 \\ \Gamma(4) &= 6 \\ \Gamma(5) &= 24 \\ \Gamma(6) &= 120 \\ \cdots \end{aligned}$ We see from above that the gamma function seems to be increasing.

True or false:

The gamma function $\Gamma(x)$ is increasing for all positive real $x$ .

False True

2 solutions

Jaydee Lucero
Jul 13, 2017

Without doing the necessary calculus for this problem, you may have recalled that $\Gamma\left(\dfrac{1}{2}\right) = \sqrt{\pi}>1$ and $\Gamma(1) = 1$ , from which it becomes evident that the gamma function is not increasing for positive $x$ . The answer is False .

Arthur Conmy
Jul 28, 2017

Alternatively, you may only know that the gamma function is an extension of the factorial function to all reals, not just the non-negative integers, and that $\Gamma(n+1)=n!$ . So $\Gamma(1)=0!=1$ and $\Gamma(2)=1!=1$ . Intuitively, a function cannot be always increasing in an interval with equal bounds $^1$ , so the answer is false (see image).

$^1$ We can use the intermediate value theorem, assuming $\Gamma$ is 'nice' in $(1, 2)$ to prove the existence of a critical point, also showing the answer to be false.

Γ ( problem ) \Gamma(\text{problem}) Γ ( problem )

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$\Gamma(\text{problem})$