Skip to main content

Illustration Shifted delta function picks out a function value

Shifted delta function picks out a function value
Get illustration

Share — copy and redistribute the material in any medium or format

Adapt — remix, transform, and build upon the material for any purpose, even commercially.

Sharing and adapting of the illustration is allowed with indication of the link to the illustration.

The shifted Dirac's delta function \( \delta(x - x_0)\) picks in an integral the function value \(f(0)\) at the position \( x = x_0 \):$$ \int_{-\infty}^{\infty} f(x) \, \delta(x-x_0) ~ \text{d}x ~=~ f(x_0) $$

If the integral is considered in the interval between \(a\) and \(b\):$$ \int_{a}^{b} f(x) \, \delta(x-x_0) ~ \text{d}x ~=~ f(x_0) $$then \(x=x_0\) must lie in this interval, otherwise the integral is zero.