Gamma Function
Extends factorials to real numbers.
Key idea
Extends factorials to real numbers.
In depth
Γ(z) = ∫₀^∞ t^(z−1)e⁻ᵗ dt extends the factorial: Γ(n) = (n − 1)! for integers. It satisfies Γ(z + 1) = zΓ(z), and Γ(1/2) = √π.
Worked example
Γ(5) = 4! = 24. Γ(3/2) = (1/2)Γ(1/2) = √π/2.
Formula
Real-world uses
- Generalising factorials for use in probability distributions and physics formulas
Common mistake
Shifting by one: Γ(n) = (n − 1)!, not n!.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Gamma Function?
Extends factorials to real numbers. It is a topic at the expert level in Calculus.
What is the formula for Gamma Function?
The key formula is Γ(n) = ∫₀^∞ tⁿ⁻¹e⁻ᵗ dt.
Where is Gamma Function used in real life?
Real-world examples: Generalising factorials for use in probability distributions and physics formulas.
What should I learn before Gamma Function?
Make sure you are comfortable with Differentiating Under the Integral, the previous topic in Calculus.
What is a common mistake with Gamma Function?
Shifting by one: Γ(n) = (n − 1)!, not n!.
How can I practise Gamma Function?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Gamma Function?
Move on to Beta Function next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.