Casio Function Calculations

Gamma Function:

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

Gamma Function Result:

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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 numbers. 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:

\[ \Gamma(z) = \sqrt{2\pi} \left(z + g + \frac{1}{2}\right)^{z+\frac{1}{2}} e^{-\left(z + g + \frac{1}{2}\right)} A_g(z) \]

Where:

\( z \) — Input value
\( g \) — Lanczos parameter (typically 7)
\( A_g(z) \) — Series coefficient

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

3. Applications of the 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 uses reflection formula Γ(z)Γ(1-z) = π/sin(πz).

5. Frequently Asked Questions (FAQ)

Q1: What is Γ(1)?
A: Γ(1) = 1. This follows from 0! = 1.

Q2: What is Γ(0.5)?
A: Γ(0.5) = √π ≈ 1.77245385091.

Q3: Can I compute negative values?
A: Yes, but not for negative integers (where it has poles). The calculator uses reflection formula for negative inputs.

Q4: How accurate is this calculator?
A: The Lanczos approximation with g=7 provides about 15 decimal digits of accuracy.

Q5: What's the relationship to factorial?
A: For positive integers n, Γ(n) = (n-1)!. So 5! = Γ(6) = 120.