-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 onedimensional 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 convenient 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)
showing that F is independent of the angular variable 8. Expression(A 11)is termed the Fourier-Bessel transform of f. The reader can easily verify that f can be recovered from F through f(p,x3,…,xN)=F(p,x3,…,xN)J(pp)Pdp the inverse fourier-Bessel transform A review of complex contour integration Some powerful techniques for the evaluation of integrals rest on complex variable the- ory. In particular, the computation of the Fourier inversion integral is often aided by these techniques. We therefore provide a brief review of this material. For a fuller discussion the reader may refer to one of many widely available textbooks on complex We shall denote by f(z) a complex valued function of a complex variable z. That f(z)=u(x, y)+ ju(r, y) where the real and imaginary parts u(x, y) and v(x, y) of f are each functions of the real and imaginary parts x and y of z z=x+jy= Re(z)+j Im(z) Herej=v-1, as is mostly standard in the electrical engineering literature. Limits, differentiation, and analyticity. Let w= f(z), and let z0= xo jyo and wo= lo+ jvo be points in the complex z and w planes, respectively. We say that wo the limit of f(z)as z approaches zo, and write m f(z)=wo if and only if both u(x,y)→ uo and u(x,y)→ Uo as x→ xo and y→ yo independently The derivative of f(z)at a point z= zo is defined by the limit f()=lim f()-f(zo if it exists. Existence requires that the derivative be independent of direction of approach that is, f(zo) cannot depend on the manner in which z zo in the complex plane(This urns out to be a much stronger condition than simply requiring that the functions u and u be differentiable with respect to the variables x and y. We say that f(z)is analytic at zo if it is differentiable at zo and at all points in some neighborhood of zo If f(z) is not analytic at zo but every neighborhood of zo contains a point at which f(z) is analytic, then zo is called a singular point of f(z) Laurent expansions and residues. Although Taylor series can be used to expand complex functions around points of analyticity, we must often expand functions around points zo at or near which the functions fail to be analytic. For this we use the Laurent 0 2001 by CRC Press LLC
showing that F is independent of the angular variable θ. Expression (A.11) is termed the Fourier–Bessel transform of f . The reader can easily verify that f can be recovered from F through f (ρ, x3,..., xN ) = ∞ 0 F(p, x3,..., xN )J0(ρp) pdp, the inverse Fourier–Bessel transform. A review of complexcontour integration Some powerful techniques for the evaluation of integrals rest on complex variable theory. In particular, the computation of the Fourier inversion integral is often aided by these techniques. We therefore provide a brief review of this material. For a fuller discussion the reader may refer to one of many widely available textbooks on complex analysis. We shall denote by f (z) a complex valued function of a complex variable z. That is, f (z) = u(x, y) + jv(x, y), where the real and imaginary parts u(x, y) and v(x, y) of f are each functions of the real and imaginary parts x and y of z: z = x + jy = Re(z) + j Im(z). Here j = √−1, as is mostly standard in the electrical engineering literature. Limits, differentiation, and analyticity. Let w = f (z), and let z0 = x0 + jy0 and w0 = u0 + jv0 be points in the complex z and w planes, respectively. We say that w0 is the limit of f (z) as z approaches z0, and write lim z→z0 f (z) = w0, if and only if both u(x, y) → u0 and v(x, y) → v0 as x → x0 and y → y0 independently. The derivative of f (z) at a point z = z0 is defined by the limit f (z0) = lim z→z0 f (z) − f (z0) z − z0 , if it exists. Existence requires that the derivative be independent of direction of approach; that is, f (z0) cannot depend on the manner in which z → z0 in the complex plane. (This turns out to be a much stronger condition than simply requiring that the functions u and v be differentiable with respect to the variables x and y.) We say that f (z) is analytic at z0 if it is differentiable at z0 and at all points in some neighborhood of z0. If f (z) is not analytic at z0 but every neighborhood of z0 contains a point at which f (z) is analytic, then z0 is called a singular point of f (z). Laurent expansions and residues. Although Taylor series can be used to expand complex functions around points of analyticity, we must often expand functions around points z0 at or near which the functions fail to be analytic. For this we use the Laurent
expansion, a generalization of the Taylor expansion involving both positive and negative powers of Z-z0 = (z-30) +>an(z-Zo The numbers an are the coefficients of the Laurent expansion of f(z) at point z= zo The first series on the right is the principal part of the Laurent expansion, and the second series is the regular part. The regular part is an ordinary power series, hence it converges in some disk 1z-zol R where R>0. Putting S=1/(2-zo), the principal part becomes >"; this power series converges for IsI p where p >0, hence the principal part converges for Iz- zol >1/p=r. When r<R, the Laurent expansion converges in th annulus r<Iz-zol R; when r>R, it diverges everywhere in the complex plane. The function f(z) has an isolated singularity at point zo if f(z) is not analyt but is analytic in the "punctured disk"0< Iz-zol R for some R>0. Isolated angularities are classified by reference to the Laurent expansion. Three types can arise 1. Removable singularity. The point zo is a removable singularity of f(z)if the principal art of the Laurent expansion of f(z) about zo is identically zero (i.e if an =0 forn=-1,-2,-3,) 2. Pole of order k. The point zo is a pole of order k if the principal part of the laurent expansion about zo contains only finitely many terms that form a polynomial of egree k in(z-z0)-. A pole of order 1 is called a simple pole 3. Essential singularity. The point zo is an essential singularity of f(z) if the principal part of the Laurent expansion of f(z)about zo contains infinitely many terms (i.e if a-n+0 for infinitely many n) The coefficient a-I in the Laurent expansion of f(z) about an isolated singular point is the residue of f(z)at zo. It can be shown that a-1 f(z)dz where r is any simple closed curve oriented counterclockwise and containing in its interior zo and no other singularity of f(z). Particularly useful to us is the formula for evaluation of residues at pole singularities. If f(z) has a pole of order k at z= Zo, then the residue of f(z)at zo is given by (k-1)!z→z0 -k-1 (z-x0)f(x) (A.13) Cauchy-Goursat and residue theorems. It can be shown that if f(z) is analytic at all points on and within a simple closed contour C, then f(z)dz=0 This central result is known as the Cauchy-Goursat theorem. We shall not offer a proof, but shall proceed instead to derive a useful consequence known as the residue theorem 0 2001 by CRC Press LLC
expansion, a generalization of the Taylor expansion involving both positive and negative powers of z − z0: f (z) = ∞ n=−∞ an(z − z0) n = ∞ n=1 a−n (z − z0)n + ∞ n=0 an(z − z0) n. The numbers an are the coefficients of the Laurent expansion of f (z) at point z = z0. The first series on the right is the principal part of the Laurent expansion, and the second series is the regular part. The regular part is an ordinary power series, hence it converges in some disk |z−z0| < R where R ≥ 0. Putting ζ = 1/(z−z0), the principal part becomes ∞ n=1 a−nζ n; this power series converges for |ζ | < ρ where ρ ≥ 0, hence the principal part converges for |z − z0| > 1/ρr. When r < R, the Laurent expansion converges in the annulus r < |z − z0| < R; when r > R, it diverges everywhere in the complex plane. The function f (z) has an isolated singularity at point z0 if f (z) is not analytic at z0 but is analytic in the “punctured disk” 0 < |z − z0| < R for some R > 0. Isolated singularities are classified by reference to the Laurent expansion. Three types can arise: 1. Removable singularity. The point z0 is a removable singularity of f (z) if the principal part of the Laurent expansion of f (z) about z0 is identically zero (i.e., if an = 0 for n = −1, −2, −3,...). 2. Pole of order k. The point z0 is a pole of order k if the principal part of the Laurent expansion about z0 contains only finitely many terms that form a polynomial of degree k in (z − z0)−1. A pole of order 1is called a simple pole. 3. Essential singularity. The point z0 is an essential singularity of f (z) if the principal part of the Laurent expansion of f (z) about z0 contains infinitely many terms (i.e., if a−n = 0 for infinitely many n). The coefficient a−1 in the Laurent expansion of f (z) about an isolated singular point z0 is the residue of f (z) at z0. It can be shown that a−1 = 1 2π j f (z) dz (A.12) where is any simple closed curve oriented counterclockwise and containing in its interior z0 and no other singularity of f (z). Particularly useful to us is the formula for evaluation of residues at pole singularities. If f (z) has a pole of order k at z = z0, then the residue of f (z) at z0 is given by a−1 = 1 (k − 1)! lim z→z0 dk−1 dzk−1 [(z − z0) k f (z)]. (A.13) Cauchy–Goursat and residue theorems. It can be shown that if f (z) is analytic at all points on and within a simple closed contour C, then C f (z) dz = 0. This central result is known as the Cauchy–Goursat theorem. We shall not offer a proof, but shall proceed instead to derive a useful consequence known as the residue theorem
Figure A 1: Derivation of the residue theorem. Figure A l depicts a simple closed curve C enclosing n isolated singularities of a function f(z). We assume that f(z) is analytic on and elsewhere within C. Around each singular point zk we have drawn a circle Ck so small that it encloses no singular point other than Zk; taken together, the Ck(k=1,., n) and C form the boundary of a region in which f(z)is everywhere analytic. By the Cauchy-Goursat theorem f(z)dz+ f(z)dz=0 f(z)dz f(z)dz where now the integrations are all performed in a counterclockwise sense. By(A 12) f(z)dz=2mj∑k (A.14) where rI,..., In are the residues of f(z) at the singularities within C Contour deformat Suppose f is analytic in a region D and r is a simple closed curve in D. If r can be continuously deformed to another simple closed curve r passing out of D, then ∫(z)dz=f(x)d (A.15) To see this, consider Figure A2 where we have introduced another set of curves +y; these new curves are assumed parallel and infinitesimally close to each other. Let c be the composite curve consisting of r, +y, - and -y, in that order. Since f is analytic f(z)dz=f(z)dz+ f(z)dz+ f(z)dz+f(z)dz=0 But -r, f(a)dz=-fr f(a)dz and _ y f(z)dz=-+y f(z)dz, hence(A15) The contour deformation principle often permits us to replace an integration contour by one that is more convenient 0 2001 by CRC Press LLC
Figure A.1: Derivation of the residue theorem. Figure A.1 depicts a simple closed curve C enclosing n isolated singularities of a function f (z). We assume that f (z) is analytic on and elsewhere within C. Around each singular point zk we have drawn a circle Ck so small that it encloses no singular point other than zk ; taken together, the Ck (k = 1,..., n) and C form the boundary of a region in which f (z) is everywhere analytic. By the Cauchy–Goursat theorem C f (z) dz + n k=1 Ck f (z) dz = 0. Hence 1 2π j C f (z) dz = n k=1 1 2π j Ck f (z) dz, where now the integrations are all performed in a counterclockwise sense. By (A.12) C f (z) dz = 2π j n k=1 rk (A.14) where r1,...,rn are the residues of f (z) at the singularities within C. Contour deformation. Suppose f is analytic in a region D and is a simple closed curve in D. If can be continuously deformed to another simple closed curve without passing out of D, then f (z) dz = f (z) dz. (A.15) To see this, consider Figure A.2 where we have introduced another set of curves ±γ ; these new curves are assumed parallel and infinitesimally close to each other. Let C be the composite curve consisting of , +γ , − , and −γ , in that order. Since f is analytic on and within C, we have C f (z) dz = f (z) dz + +γ f (z) dz + − f (z) dz + −γ f (z) dz = 0. But − f (z) dz = − f (z) dz and −γ f (z) dz = − +γ f (z) dz, hence (A.15) follows. The contour deformation principle often permits us to replace an integration contour by one that is more convenient.
Figure A 2: Derivation of the contour deformation principle Principal value integrals. We must occasionally carry out integrations of the form where f(x) has a finite number of singularities xk(k=1,., n)along the real axis. Such one singularity present at point x1, for instance, we detine improper integral. With just singularities in the integrand force us to interpret I as ar →0 provided that both limits exist. When both limits do not exist, we may still be able to obtain a well-defined result by computing f(x)dx+f(r)d (i.e by taking n= E so that the limits are "symmetric"). This quantity is called the Cauchy principal value of I and is denoted f(r) P V f(x)dx+ f(r)dx+ f(x)d. In a large class of problems f(z)(i.e, f(x) with x replaced by the complex variable is analytic everywhere except for the presence of finitely many simple poles. Some of these may lie on the real axis(at points xI and some may not Consider now the integration contour C shown in Figure A.3. We choose R so large and E so small that C encloses all the poles of f that lie in the upper half of the complex 0 2001 by CRC Press LLC
Figure A.2: Derivation of the contour deformation principle. Principal value integrals. We must occasionally carry out integrations of the form I = ∞ −∞ f (x) dx where f (x) has a finite number of singularities xk (k = 1,..., n) along the real axis. Such singularities in the integrand force us to interpret I as an improper integral. With just one singularity present at point x1, for instance, we define ∞ −∞ f (x) dx = lim ε→0 x1−ε −∞ f (x) dx + lim η→0 ∞ x1+η f (x) dx provided that both limits exist. When both limits do not exist, we may still be able to obtain a well-defined result by computing lim ε→0 x1−ε −∞ f (x) dx + ∞ x1+ε f (x) dx (i.e., by taking η = ε so that the limits are “symmetric”). This quantity is called the Cauchy principal value of I and is denoted P.V. ∞ −∞ f (x) dx. More generally, we have P.V. ∞ −∞ f (x) dx = lim ε→0 x1−ε −∞ f (x) dx + x2−ε x1+ε f (x) dx + +···+ xn−ε xn−1+ε f (x) dx + ∞ xn+ε f (x) dx for n singularities x1 < ··· < xn. In a large class of problems f (z) (i.e., f (x) with x replaced by the complex variable z) is analytic everywhere except for the presence of finitely many simple poles. Some of these may lie on the real axis (at points x1 < · · · < xn, say), and some may not. Consider now the integration contour C shown in Figure A.3. We choose R so large and ε so small that C encloses all the poles of f that lie in the upper half of the complex