Equations · Calculus and analysis
Gaussian integral
The area under the bell curve e^(−x²) over the whole real line is exactly the square root of pi.
∫₋∞^∞ e^(−x²) dx = √π
Aethel says
the integral from negative infinity to infinity of e to the power of negative x squared, d x equals the square root of pi
What it says
The function e to the power of minus x squared has no antiderivative in elementary terms, yet its total area is known exactly. The classic proof squares the integral and switches to polar coordinates.
This constant is why the normal distribution's formula carries a square root of 2π: it makes the total probability equal to one.
Symbol by symbol
| Symbol | Say it | Means |
|---|---|---|
| ∫₋∞^∞ | the integral from negative infinity to infinity of | the integral over the whole real line |
| e^(−x²) | e to the power of negative x squared | the Gaussian function, a bell-shaped curve |
| √π | the square root of pi | about 1.77245 |
Where you meet it
Probability and statistics, quantum mechanics and heat diffusion.
Questions
How do you say Gaussian integral out loud?
∫₋∞^∞ e^(−x²) dx = √π is read: “the integral from negative infinity to infinity of e to the power of negative x squared, d x equals the square root of pi.” That is exactly what Aethel says when it meets the formula in a book.
What does ∫₋∞^∞ mean in Gaussian integral?
∫₋∞^∞ (said “the integral from negative infinity to infinity of”) is the integral over the whole real line.
What does e^(−x²) mean in Gaussian integral?
e^(−x²) (said “e to the power of negative x squared”) is the Gaussian function, a bell-shaped curve.
Can I listen to the equations in my own textbook like this?
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