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 <0 2. Signum function 0, 0. 3. Rectangular pulse function 0.|x>1 4. Triangular pulse function △(x)J1-1x,xl<1 x|>1 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 LLCfor 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, 0, x > 0. (A.5) 2. Signum function sgn(x) =  −1, x < 0, 1, x > 0. (A.6) 3. Rectangular pulse function rect(x) =  1, |x| < 1, 0, |x| > 1. (A.7) 4. Triangular pulse function (x) =  1 − |x|, |x| < 1, 0, |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 appli￾cations 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