Appendix A Mathematical appendix A.1 The fourier transform The Fourier transform permits us to decompose a complicated field structure into elemental components. This can simplify the computation of fields and provide physical insight into their spatiotemporal behavior. In this section we review the properties of the transform and demonstrate its usefulness in solving field equation One-dimensional case Let f be a function of a single variable x. The Fourier transform of f(r)is the function F(k) defined by the integral Flf())=F()=f(r)e-jkx dx Note that x and the corresponding transform variable k must have reciprocal units: if x is time in seconds, then k is a temporal frequency in radians per second; if x is a length in meters, then k is a spatial frequency in radians per meter. We sometimes refer to F(k) as the frequency spectrum of f(x) Not every function has a outer tl ansform. The existence of(A 1)can be guaranteed by a set of sufficient conditions such as the following: 1. f is absolutely integrable: /_oo If(x)ldx <oo 2. f has no infinite discontinuities 3. f has at most finitely many discontinuities and finitely many extrema in any finite interval(a, b) While such rigor is certainly of mathematical value, it may be of less ultimate use to the engineer than the following heuristic observation offered by Bracewell 22 mathematical model of a physical process should be Fourier transformable. That is, if the Fourier transform of a mathematical model does not exist, the model cannot precisely describe a physical process. The usefulness of the transform hinges on our ability to recover f through the inverse F{F(k)}=f(x)= (A.2) 0 2001 by CRC Press LLC
Appendix A Mathematical appendix A.1 The Fourier transform The Fourier transform permits us to decompose a complicated field structure into elemental components. This can simplify the computation of fields and provide physical insight into their spatiotemporal behavior. In this section we review the properties of the transform and demonstrate its usefulness in solving field equations. One-dimensional case Let f be a function of a single variable x. The Fourier transform of f (x) is the function F(k) defined by the integral F{ f (x)} = F(k) = ∞ −∞ f (x)e− jkx dx. (A.1) Note that x and the corresponding transform variable k must have reciprocal units: if x is time in seconds, then k is a temporal frequency in radians per second; if x is a length in meters, then k is a spatial frequency in radians per meter. We sometimes refer to F(k) as the frequency spectrum of f (x). Not every function has a Fourier transform. The existence of (A.1) can be guaranteed by a set of sufficient conditions such as the following: 1. f is absolutely integrable: ∞ −∞ | f (x)| dx < ∞; 2. f has no infinite discontinuities; 3. f has at most finitely many discontinuities and finitely many extrema in any finite interval (a, b). While such rigor is certainly of mathematical value, it may be of less ultimate use to the engineer than the following heuristic observation offered by Bracewell [22]: a good mathematical model of a physical process should be Fourier transformable. That is, if the Fourier transform of a mathematical model does not exist, the model cannot precisely describe a physical process. The usefulness of the transform hinges on our ability to recover f through the inverse transform: F−1 {F(k)} = f (x) = 1 2π ∞ −∞ F(k) e jkx dk. (A.2)
When this is possible we write f(x)分F(k) and say that f(x) and F(k) form a Fourier transform pair. The Fourier integral theorem states that FFIf(x))=F- FIf(x)= f(r). except at points of discontinuity of f. At a jump discontinuity the inversion formula returns the average value of the one-sided limits f(x+) and f(x")of f(x). At points of continuity the forward and inverse transforms are unique Transform theorems and properties. We now review some basic facts pertaining to the Fourier transform. Let f(x)+ F()=R()+jX(), and g(x)+ G() 1. Linearity. af(x)+ Bg(x)* aF(k)+ BG(k) if a and B are arbitrary constants This follows directly from the linearity of the transform integral, and makes the transform useful for solving linear differential equations(e. g, Maxwells equations) 2. Symmetry. The property F(x)<>2f(k) is helpful when interpreting transform ables in which transforms are listed only in the forward direction 3. Conjugate function. We have f(x)+F(k) 4. Real function. If f is real, then F(k)=F*(k). Also, R(k)=/f(r)coskxdx, X()=-/f(x)sinkxdx f()=Re/ F(k)/ dk A real function is completely determined by its positive frequency spectrum. It obviously advantageous to know this when planning to collect spectral data 5. Real function with reflection symmetry. If f is real and even, then X()=0 and R(k)=2 f(x)cos kx dx, f(x) R(k)cos kx dk If f is real and odd, then R()=0 an X(k)=-2/f(x) X()sin kx dk (Recall that f is even if f(x)=f(x)for all x. Similarly f is odd if f(x)=-f(x) 6. Causal function. Recall that f is causal if f(x)=0 for x <0 (a)If f is real and causal, then x(=-2C 0 2001 by CRC Press LLC
When this is possible we write f (x) ↔ F(k) and say that f (x) and F(k) form a Fourier transform pair. The Fourier integral theorem states that F F−1 { f (x)} = F−1 F{ f (x)} = f (x), except at points of discontinuity of f . At a jump discontinuity the inversion formula returns the average value of the one-sided limits f (x+) and f (x−) of f (x). At points of continuity the forward and inverse transforms are unique. Transform theorems and properties. We now review some basic facts pertaining to the Fourier transform. Let f (x) ↔ F(k) = R(k) + j X(k), and g(x) ↔ G(k). 1. Linearity. αf (x) + βg(x) ↔ αF(k) + βG(k) if α and β are arbitrary constants. This follows directly from the linearity of the transform integral, and makes the transform useful for solving linear differential equations (e.g., Maxwell’s equations). 2. Symmetry. The property F(x) ↔ 2π f (−k) is helpful when interpreting transform tables in which transforms are listed only in the forward direction. 3. Conjugate function. We have f ∗(x) ↔ F∗(−k). 4. Real function. If f is real, then F(−k) = F∗(k). Also, R(k) = ∞ −∞ f (x) cos kx dx, X(k) = − ∞ −∞ f (x)sin kx dx, and f (x) = 1 π Re ∞ 0 F(k)e jkx dk. A real function is completely determined by its positive frequency spectrum. It is obviously advantageous to know this when planning to collect spectral data. 5. Real function with reflection symmetry. If f is real and even, then X(k) ≡ 0 and R(k) = 2 ∞ 0 f (x) cos kx dx, f (x) = 1 π ∞ 0 R(k) cos kx dk. If f is real and odd, then R(k) ≡ 0 and X(k) = −2 ∞ 0 f (x)sin kx dx, f (x) = − 1 π ∞ 0 X(k)sin kx dk. (Recall that f is even if f (−x) = f (x) for all x. Similarly f is odd if f (−x) = − f (x) for all x.) 6. Causal function. Recall that f is causal if f (x) = 0 for x < 0. (a) If f is real and causal, then X(k) = − 2 π ∞ 0 ∞ 0 R(k ) cos k x sin kx dk dx, R(k) = − 2 π ∞ 0 ∞ 0 X(k )sin k x cos kx dk dx.
(b)If f is real and causal, and f(O) is finite, then R(k) and X(k) are related by the Hilbert transforms X() R(k) dk, R(=-Pv X(k) (c)If f is causal and has finite energy, it is not possible to have F(k)=0 for k I t1. If f()=0 for both t I2, then it is not possible to have F()=0 for both k kI and k > k2 where k2>k1. That is, a time-limited signal cannot be band-limited. Similarly, a band-limited signal cannot be time-limited 8. Null function. If the forward or inverse transform of a function is identically zero, then the function is identically zero. This important consequence of the Fourier tegral theorem is useful when solving homogeneous partial differential equations n the frequency domain. 9. Space or time shift. For any fixed xo f(x-xo)+ F(k)- Jxta. a temporal or spatial shift affects only the phase of the transform, not the magni- tude 10. Frequency shift. For any fixed ko f(r)e Note that if f < F where f is real, then frequency-shifting F causes f to be- come complex -again, this is important if F has been obtained experimentally through computation in the fre 11. Similarity. We have where a is any real constant. " Reciprocal spreading" is exhibited by the Fourier transform pair; dilation in space or time results in compression in frequency, and 12. Convolution. We have fi(rf2(x-x)+ Fi(k) F2(k) f1(x)f2(x)分 Fc)F2(k-k)di 0 2001 by CRC Press LLC
(b) If f is real and causal, and f (0) is finite, then R(k) and X(k) are related by the Hilbert transforms X(k) = − 1 π P.V. ∞ −∞ R(k) k − k dk , R(k) = 1 π P.V. ∞ −∞ X(k) k − k dk . (c) If f is causal and has finite energy, it is not possible to have F(k) = 0 for k1 t1. If f (t) = 0 for both t t2, then it is not possible to have F(k) = 0 for both k k2 where k2 > k1. That is, a time-limited signal cannot be band-limited. Similarly, a band-limited signal cannot be time-limited. 8. Null function. If the forward or inverse transform of a function is identically zero, then the function is identically zero. This important consequence of the Fourier integral theorem is useful when solving homogeneous partial differential equations in the frequency domain. 9. Space or time shift. For any fixed x0, f (x − x0) ↔ F(k)e− jkx0 . (A.3) A temporal or spatial shift affects only the phase of the transform, not the magnitude. 10. Frequency shift. For any fixed k0, f (x)e jk0 x ↔ F(k − k0). Note that if f ↔ F where f is real, then frequency-shifting F causes f to become complex — again, this is important if F has been obtained experimentally or through computation in the frequency domain. 11. Similarity. We have f (αx) ↔ 1 |α| F k α , where α is any real constant. “Reciprocal spreading” is exhibited by the Fourier transform pair; dilation in space or time results in compression in frequency, and vice versa. 12. Convolution. We have ∞ −∞ f1(x ) f2(x − x ) dx ↔ F1(k)F2(k) and f1(x) f2(x) ↔ 1 2π ∞ −∞ F1(k )F2(k − k ) dk .
The first of these is particularly useful when a problem has been solved in the frequency domain and the solution is found to be a product of two or more functions of k 13. Parseval's identity. We have If(x)- dx IF()Idk Computations of energy in the time and frequency domains always give the same 14. Differentiation. We have df(x) *(k)"F() and (-jx)"f(x)<> d"F(k) dxn The Fourier transform can convert a differential equation in the x domain into an algebraic equation in the k domain, and vice versa 15. Integration. We have f(u)dn分丌F(k)(k)+ F(k) where &()is the Dirac delta or unit impulse Generalized Fourier transforms and distributions. It is worth noting that many useful functions are not Fourier transformable in the sense given above. An example the signum function 1,x<0, Although this function lacks a Fourier transform in the usual sense, for practical purposes it may still be safely associated with what is known as a generalized Fourier transform. a treatment of this notion would be out of place here; however, the reader should certainly be prepared to encounter an entry such as sgn(x)分2/jk in a standard Fourier transform table. Other functions can be regarded as possessing transforms when generalized functions are permitted into the discussion. An important example of a generalized function is the Dirac delta &(x), which has enormous value in describing distributions that are very thin, such as the charge layers often found on conductor surfaces. We shall not delve into the intricacies of distribution theory However, we can hardly avoid dealing with generalized functions: to see this we need look no further than the simple function cos kox with its transform pair cos kox + S(k+ko)+&(k- ko) The reader of this book must therefore know the standard facts about S(x): that it cquires meaning only as part of an integrand, and that it satisfies the sifting property 8(x-xo)f(x)dx= f(xo) 0 2001 by CRC Press LLC
The first of these is particularly useful when a problem has been solved in the frequency domain and the solution is found to be a product of two or more functions of k. 13. Parseval’s identity. We have ∞ −∞ | f (x)| 2 dx = 1 2π ∞ −∞ |F(k)| 2 dk. Computations of energy in the time and frequency domains always give the same result. 14. Differentiation. We have dn f (x) dx n ↔ (jk) nF(k) and (− j x) n f (x) ↔ dnF(k) dkn . The Fourier transform can convert a differential equation in the x domain into an algebraic equation in the k domain, and vice versa. 15. Integration. We have x −∞ f (u) du ↔ π F(k)δ(k) + F(k) jk where δ(k) is the Dirac delta or unit impulse. Generalized Fourier transforms and distributions. It is worth noting that many useful functions are not Fourier transformable in the sense given above. An example is the signum function sgn(x) = −1, x 0. Although this function lacks a Fourier transform in the usual sense, for practical purposes it may still be safely associated with what is known as a generalized Fourier transform. A treatment of this notion would be out of place here; however, the reader should certainly be prepared to encounter an entry such as sgn(x) ↔ 2/jk in a standard Fourier transform table. Other functions can be regarded as possessing transforms when generalized functions are permitted into the discussion. An important example of a generalized function is the Dirac delta δ(x), which has enormous value in describing distributions that are very thin, such as the charge layers often found on conductor surfaces. We shall not delve into the intricacies of distribution theory. However, we can hardly avoid dealing with generalized functions; to see this we need look no further than the simple function cos k0x with its transform pair cos k0x ↔ π[δ(k + k0) + δ(k − k0)]. The reader of this book must therefore know the standard facts about δ(x): that it acquires meaning only as part of an integrand, and that it satisfies the sifting property ∞ −∞ δ(x − x0) f (x) dx = f (x0)
for any continuous function f. With f(x)=l we obtain the familiar relation 8(x)dx=1. With f(x)=e-jkr we obtain 8(x)e y thus 8(x)+1 It follows that Useful transform pairs. Some of the more common Fourier transforms that arise in the study of electromagnetics are given in Appendix C. These often involve the simple functions defined here 1. Unit step function 1 4. Triangular pulse function △(x)J1-1x,xl1 5. Sinc function Transforms of multi-variable functions Fourier transformations can be performed over multiple variables by successive appli- cations of(A 1). For example, the two-dimensional Fourier transform over xI and x] of the function f (x1, x2, x3, .. xN) is the quantity F(kx, kx, x3,..., xN) given by f(x1, x2,x xn)e-jka dxi e-jkz2 2 dx2 0 2001 by CRC Press LLC
for any continuous function f . With f (x) = 1 we obtain the familiar relation ∞ −∞ δ(x) dx = 1. With f (x) = e− jkx we obtain ∞ −∞ δ(x)e− jkx dx = 1, thus δ(x) ↔ 1. It follows that 1 2π ∞ −∞ e jkx dk = δ(x). (A.4) Useful transform pairs. Some of the more common Fourier transforms that arise in the study of electromagnetics are given in Appendix C. These often involve the simple functions defined here: 1. Unit step function U(x) = 1, x 0. (A.5) 2. Signum function sgn(x) = −1, x 0. (A.6) 3. Rectangular pulse function rect(x) = 1, |x| 1. (A.7) 4. Triangular pulse function (x) = 1 − |x|, |x| 1. (A.8) 5. Sinc function sinc(x) = sin x x . (A.9) Transforms of multi-variable functions Fourier transformations can be performed over multiple variables by successive applications of (A.1). For example, the two-dimensional Fourier transform over x1 and x2 of the function f (x1, x2, x3,..., xN ) is the quantity F(kx1 , kx2 , x3,..., xN ) given by ∞ −∞ ∞ −∞ f (x1, x2, x3,..., xN ) e− jkx1 x1 dx1 e− jkx2 x2 dx2
-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 rR, 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"00. 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| 1/ρr. 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 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
