fx-991 CW: Gamma Function

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.

Casio fx-991CW: Input Output Formats

Casio fx-991CW: Input Output Formats

The Four Formats

There are four input/output formats that the Casio fx-991CW has. With each mode, I will show three example problems and the default format the of the output. If you always want Decimal approximations, I would suggest setting the calculator to either MathI/DecimalO or LineI/DecimalO settings.

To set the input/output:

1. Press [ SETTINGS ].

2. Select Calc Settings and press either [ EXE ], [ OK ], or [ → ].

3. Select Input/Output and press either [ EXE ], [ OK ], or [ → ].

4. Select a format and press either [ EXE ] or [ OK ]. The circle next to the mode selected will be filled.

5. To quickly return the home screen, press [ AC ].

MathI/MathO: Math Input/Math Output

Expressions are entered and displayed as written in textbook fashion. The fx-991CW returns results in exact form whenever possible in terms of π, square roots, and fractions. To get decimal approximations, press [ FORMAT ] and select Decimal.

Default (Radians Mode):

4 / 5 + 3 / 8 returns 47 / 40

√440 returns 2 × √110

cos^-1(0.5) returns 1 / 3 × π

MathI/DecimalO: Math Input/Decimal Ouptut

Expressions are entered and displayed as written in textbook fashion. However, in this mode, the fx-991CW always returns the results as decimal expressions. You can still ask for exact answers, whenever possible, by pressing [ FORMAT ] and selecting Standard.

Default (Radians Mode):

4 / 5 + 3 / 8 returns 1.175

√440 returns 20.97617696

cos^-1(0.5) returns 1.04719551

LineI/LineO: Line Input/Line Output

Expressions are entered in one-line format, similar to classic computers and calculators. Results, when possible and when fractions are used, are returned using fractions. Otherwise, all other results are returned in decimal approximation. To get decimal approximations on fractions, press [ FORMAT ] and select Decimal.

Default (Radians Mode):

4 / 5 + 3 / 8 returns 47 / 40

√440 returns 20.97617696

cos^-1(0.5) returns 1.04719551

LineI/DecimalO: Line Input/Decimal Output

Expressions are entered in one-line format, similar to classic computers and calculators. However, in this mode, the fx-991CW always returns the results as decimal expressions. To see if the approximation can be converted to fractions (within the limitation of the calculator) press [ FORMAT ] and select Standard.

Default (Radians Mode):

4 / 5 + 3 / 8 returns 1.175

√440 returns 20.97617696

cos^-1(0.5) returns 1.04719551

I hope you find this helpful. Until next time,

Eddie

All original content copyright, © 2011-2024. 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.