Proof of inverse Laplace transform

Question

Why is

$$f(t) = \frac{1}{2πj}\int_{\sigma-j\infty}^{\sigma+j\infty} F(s) e^{st} \, ds,$$

provided that

$$F(s) = \int_{0}^{\infty} f(t) e^{-st} \, dt \ ?$$

I tried to find out myself, or searched online and found a term Bromwich integral, but I want to know how this expression is derived. (And I couldn't find any :()

Thank you.

Answer 1

It is the Fourier inversion formula in disguise. In case you have never encountered this theorem before, let me prove the following version (which is obviously far from optimal).

Proposition. Let $F(s) = \int_{0}^{\infty} f(t)e^{-st} \, dt$ be the Laplace transform of $f : [0,\infty) \to \mathbb{R}$. Assume that the following technical conditions hold with some $g : [0,\infty) \to \mathbb{R}$ and $\sigma \in \mathbb{R}$:

• $f(t) = f(0) + \int_{0}^{t} g(u) \, du$. (In particular, $g$ is the 'derivative' of $f$.)
• Both $f(t)e^{-\sigma t}$ and $g(t)e^{-\sigma t}$ are Lebesgue-integrable on $[0, \infty)$.

Then for any $s > 0$, we have $$ \lim_{R\to\infty} \frac{1}{2\pi i} \int_{\sigma-iR}^{\sigma+iR} F(z)e^{s z} \, dz = f(s). $$

Proof. Define $S(x) = \frac{1}{2} + \frac{1}{\pi}\int_{0}^{x} \frac{\sin t}{t} \, dt$. Then $S(x)$ is bounded, and by Dirichlet integral, we have

$$ \lim_{R\to\infty} S(Rx) = H(x) := \begin{cases} 1, & x > 0 \\ \frac{1}{2}, & x = 0 \\ 0, & x < 0 \end{cases} $$

(Obviously $H$ denotes the Heaviside step function.) Now we have

\begin{align*} \frac{1}{2\pi i} \int_{\sigma-iR}^{\sigma+iR} F(z)e^{s z} \, dz &= \frac{1}{2\pi} \int_{-R}^{R} F(\sigma + i\xi)e^{s(\sigma+i\xi)} \, d\xi \\ &= \frac{1}{2\pi} \int_{-R}^{R} \left( \int_{0}^{\infty} f(t)e^{-(\sigma+i\xi)t} \, dt \right)e^{s(\sigma+i\xi)} \, d\xi. \end{align*}

By Fubini's theorem, we can interchange the order of integral to obtain

\begin{align*} \frac{1}{2\pi i} \int_{\sigma-iR}^{\sigma+iR} F(z)e^{s z} \, dz &= \int_{0}^{\infty} f(t)e^{-(t-s)\sigma} \left( \frac{1}{2\pi} \int_{-R}^{R} e^{(s-t)i\xi} \, d\xi \right) \, dt \\ &= \int_{0}^{\infty} f(t)e^{-(t-s)\sigma} \left( \frac{\sin R(t-s)}{\pi (t-s)} \right) \, dt \end{align*}

By the assumption, both $f(t)e^{-\sigma t}$ and $(f(t)e^{-\sigma t})' = (f'(t) - \sigma f(t))e^{-\sigma t}$ are Lebesgue-integrable. In particular, this tells that $f(t)e^{-\sigma t}$ converges to $0$ as $t\to\infty$. So by integration by parts,

\begin{align*} \frac{1}{2\pi i} \int_{\sigma-iR}^{\sigma+iR} F(z)e^{s z} \, dz &= - f(0)e^{s\sigma} S(-Rs) - \int_{0}^{\infty} (f(t)e^{-(t-s)\sigma})' S(R(t-s)) \, dt. \end{align*}

As $R \to \infty$, the right-hand side converges to

\begin{align*} \lim_{R\to\infty} \frac{1}{2\pi i} \int_{\sigma-iR}^{\sigma+iR} F(z)e^{s z} \, dz &= - \int_{0}^{\infty} (f(t)e^{-(t-s)\sigma})' H(t-s) \, dt \\ &= - \left[ f(t)e^{-(t-s)\sigma} \right]_{t=s}^{t=\infty} = f(s). \end{align*}

(Pushing the limit inside the integral is justified by the dominated convergence theorem.)

Answer 2

Here I present another version of the inversion formula for the Laplace Transform and a proof based entirely on the Fourier transform. This version extends the version described by Sangchul Lee.

Throughout this posting

• $m$ denotes the Lebesgue measure on the real line.
• For any function $f\in L^{loc}_1((0,\infty)$, its Laplace transform is defined as $$\bar{f}(s)=\int^\infty_0e^{-st}f(t)\,dt$$ If the integral above converges absolutely for some $s>0$, then $\bar{f}$ can be extended as an analytic function of the half place $H_{s}=\{z\in\mathbb{C}:\mathfrak{R}(z)>s\}$ that is continuous along the vertical line $\mathfrak{R}(z)=s$.
• For any $g\in L_1(\mathbb{R})$, its Fourier transform is defined as $$\widehat{g}(\xi)=\int_\mathbb{R} e^{-2\pi I\xi x}g(x)\,dx$$ For any complex Borel measure (or real valued measure of total finite variation) $\mu$ on $\mathbb{R}$, its Fourier transform (or its characteristic function) is defined as $$\widehat{\mu}(\xi)=\int_\mathbb{R}e^{ix\xi}\mu(dx)$$ Notice that if $\mu\ll m$ and $\mu=g\cdot m$, then $\widehat{g}(\xi)=\widehat{\mu}(-2\pi \xi)$.

Inversion formulas for the Fourier transform are well known and have been discussed here at MSE. I will use the following version of the Fourier inversion formula.

Theorem ($L_1$ summability) Suppose $f\in L_1(m)$ and that $f$ is of bounded variation in a neighborhood of some point $x\in\mathbb{R}$. Then \begin{align} \frac{f(x-)+f(x+)}{2}=\frac{1}{2\pi}\lim_{T\rightarrow\infty}\int^T_{-T}e^{-itx} \widehat{f}(-t/2\pi)\,dt\tag{4}\label{four} \end{align} where $f(x-)=\lim_{y\nearrow x}f(y)$ and $f(x+)=\lim_{y\searrow x}f(y)$.

I provide a proof of this result at the end of my posting.

For example, if $f\in L_1(m)$ is piecewise continuously differentiable, then $f$ is local bounded variation and thus, \eqref{four} holds.

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Derivation of inversion formula for Laplace Transform: Extend $f$ to $\mathbb{R}$ by setting $f(t)=0$ for $t\leq 0$. Suppose $g_c(t)=e^{-ct}f(t)\in L_1((0,\infty),m)$ for some $c>0$. It follows that $$\widehat{g_c}(\xi)=\int^\infty_{0}e^{-(c+2\pi\xi i)t}f(t)\,dt=\overline{f}(c+2\pi \xi i)$$ Then, by the summability theorem above, $$ \frac{g_c(t-)+g_c(t_+)}{2}=\frac1{2\pi}\lim_{T\rightarrow\infty}\int^T_{-T}e^{-it\xi}\widehat{g_c}(-\xi/2\pi)\,d\xi=\frac1{2\pi}\lim_{T\rightarrow\infty}\int^T_{-T}e^{it\xi}\overline{f}(c+i\xi)\,d\xi$$ at any point $t$ around which $g_c$ (equivalently $f$) is of bounded variation. Hence, for such point $t$, $$\frac{f(t-)+f(t-)}{2}=\frac1{2\pi}\lim_{T\rightarrow\infty}\int^T_{-T}e^{t(c+i\xi)}\overline{f}(c+i\xi)\,d\xi= \frac{1}{2\pi i}\lim_{T\rightarrow\infty}\int^{c+iT}_{c-iT}e^{tz}\overline{f}(z)\,dz $$

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Proof of $L_1$ summability theorem: By Fubini's theorem we have that $$g(x):=\int^T_{-T}\widehat{f}(-t/2\pi) e^{-ixt}\,dt=\int_{\mathbb{R}}\int^T_{-T}f(y)e^{i(y-x)t}\,dt\,dy= \int_{\mathbb{R}}f(y)\frac{\sin(T(y-x))}{y-x}dy $$ Since $t\mapsto\frac{\sin t}{t}$ is even, it follows that $$g(x)=\int_{\mathbb{R}}f(y+x)\frac{\sin(Ty)}{y}dy=\int_{\mathbb{R}}f(x-y)\frac{\sin(Ty)}{y}dy$$

Suppose $f$ is of bounded variation in the interval $I_\delta=(x-\delta,x+\delta)$. With our loss of generality, we may assume that $f$ is monotone nondecreasing on $I_\delta$. Splitting the domain of integration gives \begin{align} g(x)&=\frac1\pi\int_{\mathbb{R}}\frac{f(y+x)+f(x-y)}{2}\frac{\sin Ty}{y}\,dy=\frac2\pi\int^\infty_0\frac{f(y+x)+f(x-y)}{2}\frac{\sin Ty}{y}\,dy\\ &=\frac2\pi\int^\delta_0\left(\frac{f(y+x)+f(x-y)}{2}-\frac{f(x-)+f(x+)}{2}\right)\frac{\sin Ty}{t}\,dy\\ &\quad + \frac2\pi\int^\infty_\delta\frac{f(y+x)+f(x-y)}{2}\frac{\sin Ty}{y}\,dy\\ &\quad + \frac{f(x-)+f(x+)}{2}\frac{2}{\pi}\int^\delta_0\frac{\sin Ty}{y}\,dy \end{align}

The third integral in the right converges to $\frac{f(x-)+f(x+)}{2}\frac{2}{\pi}\int^\infty_0\frac{\sin u}{u}\,du=\frac{f(x-)+f(x+)}{2}$.

The second integral in the right converges to $0$ by The Riemann-Lebesgue lemma, for $y\mapsto\frac{f(y+x)-f(x-y)}{y}\mathbb{1}_{(\delta,\infty)}(y)\in L_1(m)$.

The first integral on the right requires a little extra effort. Let $A=\sup_{y>0}\Big|\int^y_0\sin\frac{\sin y}{y}\,dy\Big|$. Define \begin{align} h(y;x):=\frac{f(y+x)+f(x-y)}{2}-\frac{f(x-)+f(x+)}{2}= \frac{f(y+x)-f(x+)}{2} +\frac{f(x-y)-f(x-)}{2} \end{align}

Given $\varepsilon>0$, there is $0<\eta<\delta$ such that $f(y)-f(x_+)<\frac{\varepsilon}{2A}$ for $x<y<x+\delta$, and $f(x-)-f(y)<\frac{\varepsilon}{2A}$ for $x-\delta<y<x$.
By the second mean value theorem for integrals there are points $\xi_+, \xi_-\in(0,\eta)$ such that \begin{align} \int^\eta_0h(y;x)\frac{\sin Ty}{y}\,dy&=\frac{f(\xi_+ x)-f(x-)}{2}\int^\eta_{\xi_+}\frac{\sin Ty}{y}\,dy+\frac{f(x-\xi_-)-f(x-)}{2}\int^\eta_{\xi_-}\frac{\sin Ty}{y}\,dy \end{align} It follows that $$\Big|\int^\eta_0h(y;x)\frac{\sin Ty}{y}\,dy\Big|<\varepsilon$$ On the other hand, $y\mapsto h(y;x)\mathbb{1}_{(\eta,\delta)}(y)\in L_1(m)$ and so, $\int^\delta_\eta h(y;x)\frac{\sin Ty}{y}\,dy$ converges to $0$ by the Riemann-Lebesgue lemma.

Putting things together we have that $$\limsup_{T\rightarrow\infty}\left|\int^T_{-T}\widehat{f}(-t/2\pi) e^{-ixt}\,dy - \frac{f(x-)+f(x-)}{2}\right|\leq\varepsilon$$ As $\varepsilon>0$ is arbitrary, the conclusion of the Theorem follows.