- #1

Yegor

- 147

- 1

Does exist any formula for this integral?

Or for [tex]\int_{-\infty}^\infty e^{-ax^2+bx} dx[/tex]???

You are using an out of date browser. It may not display this or other websites correctly.

You should upgrade or use an alternative browser.

You should upgrade or use an alternative browser.

- Thread starter Yegor
- Start date

- #1

Yegor

- 147

- 1

Does exist any formula for this integral?

Or for [tex]\int_{-\infty}^\infty e^{-ax^2+bx} dx[/tex]???

- #2

mathman

Science Advisor

- 8,063

- 541

Your first integral then becomes an erf integral, which can only be done numerically. The second integral has a closed form solution - integral of Gaussian.

- #3

lurflurf

Homework Helper

- 2,452

- 148

[tex]\int_{-\infty}^\infty e^{-ax^2} dx={\sqrt{\frac{\pi}{a}}}}[/tex]Yegor said:

Does exist any formula for this integral?

Or for [tex]\int_{-\infty}^\infty e^{-ax^2+bx} dx[/tex]???

Consider your integral multiply it by a constant of the form exp(c) where c lets you conplete the square of the quadratic. Then observe

[tex]\int_{-\infty}^\infty e^{-x^2} dx=\int_{-\infty}^\infty e^{-(x+y)^2} dx

[/tex]

for any constant y

Last edited:

- #4

lurflurf

Homework Helper

- 2,452

- 148

It is true that the first will involve erf, while the second will have nicer form. Yet erf is a closed form. Also closed form verses numerical solution is kind of silly any way. log(2) is a closed form, but if you want a number you have to "do it numerically". The issue has more do do with how many function one want to define tabulate and use. The distinction between an answer erf(1) and one of sin(1) is mostly historical.mathman said:^{2}-bx=a(x-b/2a)^{2}+(b/2)^{2}/a

Your first integral then becomes an erf integral, which can only be done numerically. The second integral has a closed form solution - integral of Gaussian.

Share:

- Last Post

- Replies
- 3

- Views
- 2K

- Last Post

- Replies
- 6

- Views
- 12K

- Last Post

- Replies
- 1

- Views
- 2K

- Last Post

- Replies
- 10

- Views
- 782

- Last Post

- Replies
- 1

- Views
- 994

- Last Post

- Replies
- 1

- Views
- 6K

- Last Post

- Replies
- 7

- Views
- 2K

- Last Post

- Replies
- 1

- Views
- 835

- Last Post

- Replies
- 3

- Views
- 8K

- Last Post

- Replies
- 2

- Views
- 2K