Table 1
Summary of the contents of the toolbox.
| Function/Procedure name | Calling Sequence | Description |
|---|---|---|
| takeAlook | takeAlook(expr, tablename) | Obtain what maps to expr in tablename |
| addToTable | addToTable(procname, expr, ans) | Add expr⇔ ans to table associated with procedure, procname |
| Ekronecker | EKronecker(n, m) | Obtain Kronecker delta function i.e. δmn=1, when n=m |
| EdDirac | EdDirac(a) | Redefines Dirac function so that integral at end points is NOT halved |
| Conv | Conv(f, g, [x], 1dcartesian) | Bracket Convolution: obtain 1D convolution of f(x) and g(x) in Cartesian coordinates |
| Conv(ff, gg, [x, y], 2dcartesian) | obtain 2D convolution of ff(x) and gg(x) in Cartesian coords. | |
| Conv(pp, qq, [r, ϑ], 2dpolar) | obtain 2D convolution of pp(r, ϑ) and qq(r, ϑ) in polar coords. | |
| Conv(u, v, [ϑ], angular) | obtain angular convolution of u(ϑ) and v(ϑ) | |
| Conv(p, q, [r], radial) | obtain radial convolution of p(r) and q(r) | |
| Conv(s, w, [n], series) | obtain convolution of the infinite series, s(n) and w(n) | |
| OneDCartConv | OneDCartConv(f, g, x) | Obtain 1D convolution of f(x) and g(x) in Cartesian coordinates |
| TwoDCartConv | TwoDCartConv(ff, gg, x, y) | Obtain 2D convolution of ff(x) and gg(x) in Cartesian coords. |
| TwoDPolarConv | TwoDPolarConv(pp, qq, r, ϑ) | Obtain 2D convolution of pp(r, ϑ) and qq(r, ϑ) in polar coords. |
| RadConv | RadConv(p, q, r) | Obtain radial convolution of p(r) and q(r) |
| AngConv | AngConv(u, v, ϑ) | Obtain angular convolution of u(ϑ) and v(ϑ) |
| SerConv | SerConv(s, w, n) | Obtain convolution of the infinite series, s(n) and w(n) |
| Hankel | Hankel(p, r, s, n) | Obtain Hankel transform of p(r) (an expression or list) |
| InvHankel | InvHankel(P, s, r, n) | Obtain Inverse Hankel transform of P(s) (expression or list) |
| FS1D | FS1D (f, x, n, range, condition) | Obtain 1D Fourier Series of f(x). |
| condition = “coefficientComplex” | Returns complex Cn coefficients | |
| condition = “coefficientReal” | Returns A0, An, Bn coefficients | |
| condition = “series” | Returns full sum | |
| InvFS1D | InvFS1D(F, n, x) | Obtain Inverse 1D Fourier Series of F[n] |
| Polar2DFT | Polar2DFT(pp, r, ϑ, ρ, ψ) | Obtain 2D Polar Fourier transform of pp(r, ϑ) by applying 2πi-n*(FS + nth order HT) + iFS |
| InvPolar2DFT | InvPolar2DFT(PP, ρ, ψ, r, ϑ) | Obtain Inverse 2D Polar Fourier transform of PP(ρ, ψ) by applying (in/2π)*(FS + nth order HT) + iFS |
| DirectPolar2DFT | DirectPolar2DFT(pp, r, ϑ, ρ, ψ) | Evaluate 2D Polar Fourier transform of pp(r, ϑ) from the integral definition |
| FSH | FSH(pp, r, ϑ, ρ, ψ) | Obtain step-by-step results for 2D Polar Fourier Series-Hankel Transform of pp(r, ϑ) i.e. 2πi-n*(FS + nth order HT) |
| InvFSH | InvFSH(PP, ρ, ψ, r, ϑ) | Obtain step-by-step results for Inverse 2D Polar Fourier Series-Hankel Transform of PP(ρ, ψ) i.e. (in/2π)*nth order HT + iFS |
Figure 1
Testing the toolbox on the Fourier transform of some common functions.