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 LLCThe 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, 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)