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 LLCshowing 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 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 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