Recently, I saw the Fourier transform established by the Fourier's integral theorem in "Integral\, transforms\, for Engineers" (p 50). Specifically, the exponential form of Fourier's integral theorem is $$ f(x) = \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-isx}\int_{-\infty}^{\infty}e^{ist}f(t)\,dt\,ds $$
Then the Fourier transform are defined as \begin{align} F(s)&=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{ist}f(t)dt \\ f(t)&=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}e^{-ist}F(s)ds \end{align} However, the pair of Fourier transforms are often shown as follows \begin{align} F(s)&=\int_{-\infty}^{\infty}e^{-ist}f(t)dt \\ f(t)&=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ist}F(s)ds \end{align} Are these two forms the same essentially and why (especially for the index of Fourier transform and inverse Fourier transform)?