-jka r dx d The two-dimensional inverse transform is computed by multiple application of(. 2) recovering f(x1, x2, x3,..., xN) through the operation F(ri,kr,x3,.,xN)ejkalejknz 2 dkr, dkr Higher-dimensional transforms and inversions are done analogously Transforms of separable functions. If we are able to write f(x1,x2,x3,……,xN)=f(x1,x3,……,xN)f2(x2,x3,,xN), then successive transforms on the variables x and x? result in f(x1,x,x N)分F1(kx1,x3,……,xN)F2(k In this case a multi-variable transform can be obtained with the help of a table of one- dimensional transforms, If for instance f(x,y,z)=8(x-x)8(y-y)8(z-x then we obtain by three applications of(A1) A more compact notation for multi-dimensional functions and transforms makes use of the vector notation k=&kr ok, + ik, and r=&x +yy+iz where r is the position vector. In the example above, for instance, we could have written x)8(y-y)8(z-x)=8(r-1 F(k)= sr-re dx dy dz=e Fourier-Bessel transform. If xI and x] have the same dimensions, it may be con- venient to recast the two-dimensional Fourier transform in polar coordinates. Let xI pcos o, kx,= pcos 6, x2=psin and kx,= psin 8, where p and p are defined on(0, oo) andφand6are defined on(-丌,丌).Then F(P,6,x3,…,xN) f(p,中,x3,…,xN)e-mp(-0)pdpd.(A.10) If f is independent of o(due to rotational symmetry about an axis transverse to xI and x2), then the integral can be computed using the identity Jo(x)= e J-rcos(g-e) d Thus(A10)becomes F(p f(p,x3,……,xN)J0(pp)pdp 0 2001 by CRC Press LLC= ∞ −∞ ∞ −∞ f (x1, x2, x3,..., xN ) e− jkx1 x1 e− jkx2 x2 dx1 dx2. The two-dimensional inverse transform is computed by multiple application of (A.2), recovering f (x1, x2, x3,..., xN ) through the operation 1 (2π)2 ∞ −∞ ∞ −∞ F(kx1 , kx2 , x3,..., xN ) e jkx1 x1 e jkx2 x2 dkx1 dkx2 . Higher-dimensional transforms and inversions are done analogously. Transforms of separable functions. If we are able to write f (x1, x2, x3,..., xN ) = f1(x1, x3,..., xN ) f2(x2, x3,..., xN ), then successive transforms on the variables x1 and x2 result in f (x1, x2, x3,..., xN ) ↔ F1(kx1 , x3,..., xN )F2(kx2 , x3,..., xN ). In this case a multi-variable transform can be obtained with the help of a table of one￾dimensional transforms. If, for instance, f (x, y,z) = δ(x − x )δ(y − y )δ(z − z ), then we obtain F(kx , ky , kz) = e− jkx x e− jky y e− jkzz by three applications of (A.1). A more compact notation for multi-dimensional functions and transforms makes use of the vector notation k = xˆkx + yˆky + zˆkz and r = xˆx + yˆ y + zˆz where r is the position vector. In the example above, for instance, we could have written δ(x − x )δ(y − y )δ(z − z ) = δ(r − r ), and F(k) = ∞ −∞ ∞ −∞ ∞ −∞ δ(r − r )e− jk·r dx dy dz = e− jk·r . Fourier–Bessel transform. If x1 and x2 have the same dimensions, it may be con￾venient to recast the two-dimensional Fourier transform in polar coordinates. Let x1 = ρ cos φ, kx1 = p cos θ, x2 = ρ sin φ, and kx2 = p sin θ, where p and ρ are defined on (0,∞) and φ and θ are defined on (−π,π). Then F(p,θ, x3,..., xN ) = π −π ∞ 0 f (ρ, φ, x3,..., xN ) e− jpρ cos(φ−θ)ρ dρ dφ. (A.10) If f is independent of φ (due to rotational symmetry about an axis transverse to x1 and x2), then the φ integral can be computed using the identity J0(x) = 1 2π π −π e− j x cos(φ−θ) dφ. Thus (A.10) becomes F(p, x3,..., xN ) = 2π ∞ 0 f (ρ, x3,..., xN )J0(ρp) ρ dρ, (A.11)