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Earlier in Chapter 4 we discussed the Laplace Transform, \(\mathcal{L}\), mapping spatio-temporal signals into 2D manifolds called spacekime surfaces. The Laplace transformation is a linear operator that maps complex-valued functions of a positive real variable (e.g., time, \(t \in\mathbb{R}^+\)) to complex valued functions defined over the complex plane, \(\kappa\in\mathbb{C}\)
\[{\mathcal {L}}(f)(\kappa)=\hat{f}(\kappa)\equiv F(\kappa)=\int_{0}^{\infty} {f(t)e^{-\kappa t} dt}.\]
The Laplace transform is a special case of the more general integral transform defined for a specific kernel \(K(\cdot,\cdot)\):
\[(Tf)(z)=\int_{t_{1}}^{t_{2}}f(t)\,K(t,z) dt.\]
Under certain cases, the Meijer-G function turns out to be an effective symbolic computation strategy to compute integrals of the form: \(\int_0^{\infty}f(x)dx\). It also has really important theoretical connections to the theory of complex numbers and mathematical (integral) transformations. It is an important function class that is closed with respect to many integral transformations (e.g., Fourier, Laplace transforms) and presents many flexibility to metamodeling and enhancing model interpretability. Specifically, this prior work shows the general utilization of the Meijer-G function [1]:
\[\text{Polynomials Approximants (Taylor series)} \subset \text{Algebraic Approximants (Padé)} \subset \text{Analytic Functions (Meijer-G)}\]
Let’s explore a few of these aspects.
The Meijer-G function is defined as follows: \[\begin{equation} \begin{split} G_{p,q}^{m,n}(a_1,...a_p;b_1,b_2,...b_q|z)\equiv G_{p,q}^{m,n}\left(\begin{array}{c} a_p\\ b_q \end{array}\bigg| z\right)\equiv G_{p,q}^{m,n}\left(\begin{array}{c} \overbrace{a_1,...,a_n}^{\text{n component}};\overbrace{a_{n+1},...,a_{p}}^{\text{p-n component}}\\ \underbrace{b_1,...,b_m}_{\text{m component}};\underbrace{b_{m+1},...,b_{q}}_{\text{q-m component}} \end{array}\bigg| z\right)=\\\frac{1}{2\pi i}\int_{C}\frac{\Gamma(b_1-s)...\Gamma(b_m-s)\Gamma(1-a_1+s)...\Gamma(1-a_n+s)}{\Gamma(1-b_{m+1}+s)...\Gamma(1-b_q+s)\Gamma(a_{n+1}-s)...\Gamma(a_{p}-s)}z^sds. \end{split} \end{equation}\]
A key aspect of the Meijer-G function is its relation to the Laplace transform.
The Laplace transform for Meijer-G function of \(x^2\) type is \[\begin{equation} \int_0^{\infty} e^{-s\cdot x}G_{p,q}^{m,n}\left(\begin{array}{c} a_p\\ b_q \end{array}\bigg| wx^2\right)dx = \frac{1}{\sqrt{\pi}s}G_{p+2,q}^{m,n+2}\left(\begin{array}{c} 0,\frac{1}{2},a_p\\ b_q \end{array}\bigg| \frac{4w}{s^2}\right) \label{eq1}\tag{1} \end{equation}\] More explicitly, it is
\[\frac{1}{\sqrt{\pi}s}G_{p+2,q}^{m,n+2}\left(\begin{array}{c} \overbrace{\overbrace{0,\frac{1}{2},a_1,....,a_n}^{n+2\ components};a_{n+1},....a_{p}}^{p+2\ components} \\ b_q \end{array} \bigg| \frac{4w}{s^2}\right).\]
The cosine family of base functions belong to the general Meijer-G family:
\[\begin{equation} \cos(x) = \sqrt{\pi}G_{0,2}^{1,0}\left(\begin{array}{c} ;\\ 0;\frac{1}{2} \end{array}\bigg| \frac{x^2}{4}\right). \end{equation}\]
We can directly derive this from the definition by expanding the right hand side as follows.
\[\begin{equation} \begin{split} RHS &= \sqrt{\pi} (-\frac{1}{2\pi i})\int_{C}\frac{\prod_{j=1}^m\Gamma(b_j-s)\prod_{j=1}^n\Gamma(1-a_j+s)}{\prod_{j=m+1}^q\Gamma(1-b_{j}+s)\prod_{j=n+1}^q\Gamma(a_j-s)}z^sds\bigg|_{z=\frac{x^2}{4}}\\ &=\sqrt{\pi}(-\frac{1}{2\pi i})\int_C \frac{\Gamma(-s)}{\Gamma(1-\frac{1}{2}+s)}z^sds\bigg|_{z=\frac{x^2}{4}}=\sqrt{\pi}(-\frac{1}{2\pi i}) \underbrace{2\pi i\sum_{k=0}^{\infty}res_{k}}_{\text{residue theorem}} \end{split} \end{equation}\]
The gamma function \(\Gamma(z)\) has poles \(0,-1,-2,....\), and \(\frac{1}{\Gamma(z)}\) is entire since the gamma function has no zeros. Using the fact that only the poles from the numerator results in residues which contributes to the integration, and \(\text{res}_{z=n}\Gamma(-z)=\frac{(-1)^{n+1}}{n!}\) (the rest are scalar factors), we have \[\begin{equation} \begin{split} RHS &= \sqrt{\pi}(-\frac{1}{2\pi i})2\pi i\sum_{k=0}^{\infty}\frac{(-1)^{k+1}}{k!}\frac{1}{\Gamma(\frac{1}{2}+k)}(\frac{x^2}{4})^k=-\sum_{k=0}^{\infty}\frac{(-1)^{k+1}}{k!}x^{2k}\\&=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}x^{2k}=\cos(x) \end{split} \end{equation}\] where we have used the fact that \(\frac{k!\Gamma(\frac{1}{2}+k)4^k}{\sqrt{\pi}}=(2k)!\). Since \(k!2^k=2\cdot 4\cdot 6\cdots 2k\) and \(\frac{\Gamma(\frac{1}{2}+k)2^k}{\sqrt{\pi}}=1\cdot 3\cdot 5...\cdot (2k-1).\)
Schematically, the red contour of integration, \(\gamma\), can be extended to positive infinity on the right and the only contributions to the contour integral of \(\Gamma(-z)\) are the poles \(0,1,2,\cdots \in \mathbb{R}\).
We will demonstrate the representations of the Laplace transform of the trigonometric (cosine) basis and Bessel base functions, which are commonly used to expand most “nice” functions.
The cosine base functions can be represented via the general Meijer-G function as follows.
\(\cos(x) = \sqrt{\pi}G_{0,2}^{1,0}\left(\begin{array}{c} ;\\ 0;\frac{1}{2} \end{array}\bigg| \frac{x^2}{4}\right)\).
As the general form of Meijer-G function is closed under Laplace transform (Equ 1), we have:
\[\begin{equation} \mathcal{L}(\cos(wx))(s)=\frac{s}{w^2+s^2} \end{equation}\]
\[\begin{equation} \begin{split} \mathcal{L}(\cos(wx))(s) &= \frac{1}{\sqrt{\pi}s}\sqrt{\pi}G_{2,2}^{1,2}\left(\begin{array}{c} 0,\frac{1}{2};\\ 0;\frac{1}{2} \end{array}\bigg| \frac{4w'}{s^2}\right)\bigg|_{w'=\frac{w^2}{4}}\\ &=\frac{1}{s}\left(\frac{1}{1+x}\right)\bigg|_{x=\frac{w^2}{s^2}}=\frac{1}{s}\cdot \frac{1}{1+\frac{w^2}{s^2}}=\frac{s}{w^2+s^2} \end{split} \end{equation}\]
\[\begin{equation} {}_2^1F(\frac{1+n}{2},\frac{2+n}{2},1+n,x)=\frac{2^n(1-\sqrt{1-x})^n}{\sqrt{1-x}x^n}, \forall n\in \mathbb{Z} \end{equation}\]
The Bessel J function belong to the general Meijer-G family with \(J_n(at) = G_{0,2}^{1,0}\left(\begin{array}{c} ;\\ \frac{n}{2};-\frac{n}{2} \end{array}\bigg|\frac{a^2t^2}{4}\right), -\frac{\pi}{2}\le arg(x)\equiv at \leq \frac{\pi}{2}\).
Using the general form of Meijer-G function being closed under Laplace transform (Equ 1) \[\begin{equation} \mathcal{L}(J_n(at))(s)=\frac{(-s+\sqrt{a^2+s^2})^n}{a^n\sqrt{s^2+a^2}} \end{equation}\]
Applying Equation (1), we use \(\mathcal{L}{J_n(at)}(s)=G_{2,2}^{1,2}\left(\begin{array}{c} 0,\frac{1}{2};\\ \frac{n}{2};-\frac{n}{2} \end{array}\bigg|\frac{a^2t^2}{s^2}\right)\) by referring to mathematica or we may refer to the special case for Meijer-G tabularized at wolfram website, \[\begin{equation} \begin{split} LHS&=\frac{1}{\sqrt{\pi}s}\cdot 2^{-n}\sqrt{\pi}(\frac{a^2}{s^2})^{n/2}{}_2^1F(\frac{1+n}{2},\frac{2+n}{2},1+n,-\frac{a^2}{s^2})\\&=2^{-n}s^{-1-n}a^n\frac{2^n(1-\sqrt{1-x})^n}{\sqrt{1-x}x^n}\bigg|_{x=-\frac{a^2}{s^2}}\\&=s^{n-1}\frac{1}{a^n}\frac{(-1+\sqrt{1+\frac{a^2}{s^2}})^n}{\sqrt{1+\frac{a^2}{s^2}}}=\frac{(-s+\sqrt{a^2+s^2})^n}{a^n\sqrt{s^2+a^2}} \end{split} \end{equation}\] where we made use the previous lemma.
Let’s explore several animation examples showcasing the parallels between raw and Laplace-transformed signals by expanding (continuing) the domain of the original signal (\(\mathbb{R}^+\)) into the complex plane (\(\mathbb{C}\)) using the Meijer-G functions.
The following Mathematica code illustrates the solution for the original case on the left.
gg[w_, x_] = Sqrt[Pi]*MeijerG[{{}, {}}, {{0}, {1/2}}, (w*x)^2/4]
MeijerGcos[var1_, viewops : OptionsPattern[]] :=
Module[{plot, cosinepara},
plot = Plot3D[Re[gg[var1, a + b*I]], {a, -10, 10}, {b, -2, 2},
PlotRange -> Automatic, Mesh -> None, PlotPoints -> 500,
ColorFunction -> Function[{x, y, z}, Hue[Arg[gg[var1, x + I y]]]],
ColorFunctionScaling -> False, Axes -> True,
AxesStyle -> Arrowheads[0.01]];
cosinepara =
ParametricPlot3D[{t, 0, Cos[var1*t]}, {t, -10, 10},
PlotStyle -> {Black, Thickness[0.005]}, AxesLabel -> {x, y, z}];
Show[cosinepara, plot, Axes -> True,
AxesLabel -> {Style["Time Ordered Signal - Classical t", Bold, 12],
Style["Continued axis - y", Bold, 10],
Style["Real value", Bold, 16]},
PlotRange -> {{-10, 10}, {-2, 2}, {-2, 2}}, Evaluate[viewops]]]
ani = Table[MeijerGcos[c], {c, 1, 5, 0.3}];
anifb = Join[ani, Reverse[ani]];
Export["C:\\Users\\user\\project\\cosinefb_test.gif", anifb,
"DisplayDurations" -> 0.2,
"AnimationReptitions" -> 20,
"AnimationDirection" -> ForwardBackward, ImageSize -> 800];The corresponding Mathematica code for the transformed case on the right is below.
gtrans[w_, s_] = 1/s*MeijerG[{{0, 1/2}, {}}, {{0}, {1/2}}, w^2/(s^2)]
ani2 = Table[
Plot3D[Abs[gtrans[c, a + b*I]], {a, -5, 5}, {b, -5, 5},
PlotRange -> Automatic, Mesh -> None, PlotPoints -> 500,
PlotLabel -> Style["The Transformed Domain", Bold, 20],
AxesLabel -> {Style["Transformed x", Bold, 12],
Style["Transformed y", Bold, 12], Style["Intensity", Bold, 12]},
ColorFunction ->
Function[{x, y, z}, Hue[Arg[gtrans[c, x + I y]]]],
ColorFunctionScaling -> False, Axes -> True,
AxesStyle -> Arrowheads[0.01]], {c, 1, 5, 0.3}];
ani2fb = Join[ani2, Reverse[ani2]];
Export["C:\\Users\\meijertransformed_domain1_forback.gif", ani2fb,
"DisplayDurations" -> 0.2, "AnimationReptitions" -> 20,
"AnimationDirection" -> ForwardBackward, ImageSize -> 800];