Normalization issues in wikipedia inverse Laplace transform

Question

In wikipedia, there appears under https://en.wikipedia.org/wiki/Laplace_transform#Inverse_Laplace_transform

the normalization factor $\frac{1}{2\pi i}$.

In terry's blog https://terrytao.wordpress.com/2014/12/05/245a-supplement-2-a-little-bit-of-complex-and-fourier-analysis/#lfs

there appears (and I think he's correct) the factor $\frac{1}{\pi i}$.

A confirmation of terry being right is the way to deduce the fourier transform.

Both are talking about the one sided laplace transform.

Can someone explain what is going on?

Answer 1

The inversion formula for piecewise $C^1$ functions with bounded variation and supported on $t\ge a$ is $$\lim_{\epsilon \to 0}\frac{f(t-\epsilon)+f(t+\epsilon)}{2}= \frac1{2i\pi}\int_{\sigma-i\infty}^{\sigma+i\infty} F(s)e^{st}ds, \qquad t\in \Bbb{R},\sigma > 0$$

Answer 2

The formula in the Wikipedia article on the inverse Laplace transform has the same factor of $1/(2\pi i)$ that the formula in the cited Tao blog post does, in Tao's Corollary 37. Comments from the OP seem to concern the seeming mismatch between a $1/(\pi i)$ scaling in a formula in his Corollary 27 and his use of a $1/(2\pi i)$ scaling in his Corollary 37.

Tao is a little unclear at a crucial point in his discussion of his Cor. 37, but I see nothing wrong in principle with $2\pi i$ appearing in one equation and $\pi i$ in another, connecting different but related quantities. In summary: I see no error or contradiction in what Tao wrote, nor contradiction with the relevant Wikipedia articles.

I think the source of confusion is Tao's "By contour shifting (using (14) to handle error terms) we see that..." at the top of his proof of his Corollary 27. He does not describe how the shifting is done, and so does not make clear the exact connection between the principal value $$\lim_{T\to\infty}\int_{-T}^T \mathcal L f(c+it)dt$$ and the residue of $\mathcal L f$ at $0$.

Lucky for us the details of this process are spelled out in Ahlfors's Complex Analysis, section 5.3, "Evaluation of definite integrals". On pp.157-158 (of the 3d edition) he works out an example of integrating an analytic function with a pole just at the origin, along on a rectangular contour enclosing the origin. Of course that integral is $2\pi i$ times the residue. He deforms the contour, and gets a result he describes as "on the right-hand side we observe that one-half of the residue at $0$ has been included; this is is as if one half of the pole were counted as lying in the upper half plane". (Ahlfors's set-up is rotated $\pi/2$ from Tao's, so where Tao has right half plane, Ahlfors has the upper half plane.)

Ahlfors goes on to discuss the function $f(z)=e^{iz}/z$. The principal value is $$\text{pr.v. }\int_{-\infty}^\infty\frac {e^{ix}} x dx = \pi i$$ even though the residue of $f$ at $0$ is $1$.

That's all.