Question

The Kernel of the Laplace transform is given by
1. \(e^{-st}\)
2. \(\frac{e^{ist}}{\sqrt{2\pi}}\)
3. \(\sqrt{\frac{2}{\pi}} sin~st\)
4. \(\sqrt{\frac{2}{\pi}} cos~st\)

Answer 1

Correct Answer - Option 1 : \(e^{-st}\)

Fredholm equation of the first kind or an integral transform equation is given as:

\(g(x) = \int_{a}^{b} k(x,y)~f(y)~dy\)

The bivariate function k(x, y) is called the kernel of the integral equation.

Important examples of Integral transform includes Laplace transform and Fourier transform.

The Laplace transform is an integral transform of the form:

\(F(s) = \int_{-\infty}^{\infty} f(t)~e^{-st}dt\)

∴ The kernel of Laplace transform is given as \(\textbf e^{\textbf -st}\)