fx-991 CW: Gamma Function
All screenshots were made with Casio’s classpad.net website.
No Gamma Function? No Problem!
The gamma function is used a lot in advanced mathematics. The gamma functions with a lot of definition functions, but the most common one is for values t>0 (in particular Re(t)>0):
Γ(t) = ∫( x^(t-1) × e^(-x) dx, 0, ∞)
This integral is an improper integral and unless we have a calculator that handles infinite limit, we need to use the following:
Γ(t) =
lim ∫( x^(t-1) × e^(-x) dx, 0, w)
w → ∞
Calculators with the integral function can use the above for estimating the gamma function.
Use Basic Properties for Shortcuts
If t is a positive integer, we can use the factorial function:
Γ(t) = (t – 1)!
Example: Γ(16) = (16 – 1)! = 15! ≈ 1.31 × 10^12
If t is in the form n/2 where n is odd (i.e. 1/2 = 0.5, 3/2 = 1.5, 5/2 = 2.5, 7/2 = 3.5, etc.). we can use the product..
Γ(n / 2) = (n – 2) / 2 × (n – 4) / 2 × (n – 6) / 2 × … × 1 / 2 × √π
Example: Γ(3.5) = Γ(7 / 2) = 5 / 2 × 3 / 2 × 1 / 2 × √π = 15 / 8 × √π ≈ 3.32335097
The following pictures demonstrate the use of the above equivalency along with integral estimate:
I hope you find this helpful,
Eddie
All original content copyright, © 2011-2025. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
The author does not use AI engines and never will.