Casio Gamma Function Calculator

Gamma Function:

\[ \Gamma(x) = \int_{0}^{\infty} t^{x-1} e^{-t} dt \]

Gamma Function Value:

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1. What is the Gamma Function?

The Gamma function (Γ) is a generalization of the factorial function to complex and real number arguments. For positive integers, Γ(n) = (n-1)!. It's widely used in mathematics, physics, and engineering.

2. How Does the Calculator Work?

The calculator uses the Lanczos approximation to compute the Gamma function:

\[ \Gamma(z) = \sqrt{2\pi} \left(z + g + \frac{1}{2}\right)^{z+\frac{1}{2}} e^{-\left(z + g + \frac{1}{2}\right)} \left[c_0 + \frac{c_1}{z+1} + \frac{c_2}{z+2} + \cdots + \frac{c_n}{z+n}\right] \]

Where:

\( z \) — Input value
\( g \) — Arbitrary parameter (typically 7)
\( c_n \) — Precomputed coefficients

Explanation: The Lanczos approximation provides accurate results across the complex plane (except for non-positive integers).

3. Applications of Gamma Function

Details: The Gamma function appears in probability distributions (Gamma, Chi-squared, Student's t), fractional calculus, analytic number theory, and quantum physics.

4. Using the Calculator

Tips: Enter any positive real number. The calculator will compute Γ(x). For x ≤ 0, the function has poles at negative integers.

5. Frequently Asked Questions (FAQ)

Q1: What's the relationship between Gamma and factorial?
A: For positive integers n, Γ(n) = (n-1)!. For example, Γ(5) = 4! = 24.

Q2: What is Γ(1/2)?
A: Γ(1/2) = √π ≈ 1.77245385091. This is important in normal distribution calculations.

Q3: Can Gamma handle non-integer values?
A: Yes, Γ(x) is defined for all complex numbers except non-positive integers.

Q4: What's the difference between Gamma and incomplete Gamma?
A: The incomplete Gamma function γ(s,x) is Γ(s) with upper limit x instead of ∞.

Q5: How accurate is this calculator?
A: The Lanczos approximation provides about 15 digits of accuracy for real x > 0.