- #1
facenian
- 436
- 25
I'm reading a book on Path Integral and found this formula
[itex]\int_{-\infty}^{\infty }e^{-ax^2+bx}dx=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}} [/itex]
I Know this formula to be correct for a and b real numbers, however, the author applies this formula for a and b pure imaginary and I do not understand why this is correct. This is like saying
[itex]\int_{-\infty}^{\infty}e^{-ix^2 }dx=\int_{-\infty}^{\infty}cos(x^2) dx +i\int_{-\infty}^{\infty}sen(x^2) dx=\sqrt{\pi}[/itex]
[itex]\int_{-\infty}^{\infty }e^{-ax^2+bx}dx=\sqrt{\frac{\pi}{a}}e^{\frac{b^2}{4a}} [/itex]
I Know this formula to be correct for a and b real numbers, however, the author applies this formula for a and b pure imaginary and I do not understand why this is correct. This is like saying
[itex]\int_{-\infty}^{\infty}e^{-ix^2 }dx=\int_{-\infty}^{\infty}cos(x^2) dx +i\int_{-\infty}^{\infty}sen(x^2) dx=\sqrt{\pi}[/itex]