Tham khảo tài liệu 'handbook of mathematics for engineers and scienteists part 69', khoa học tự nhiên, toán học phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 444 Integral Transforms table Main properties of the Fourier transform No. Function Fourier transform Operation 1 af 1 x bf2 x af1 u bf2 u Linearity 2 f x a a 0 af au Scaling 3 xn f x n 1 2 . fu Differentiation of the transform 4 f L x -u f u Differentiation 5 f n x tuff u Differentiation 6 ftt f1 e f2 x - e d J - flfi Convolution For brevity we rewrite formula as follows f x F 1 f u or f x u x . . Asymmetric form of the Fourier transform. Alternative Fourier transform. 1 . Sometimes it is more convenient to define the Fourier transform by f u I f x e iux dx. J - In this case the Fourier inversion formula reads 1 C f x f u e du. 2n J_ 2 . sometimes the alternative Fourier transform is used and called merely the Fourier transform which corresponds to the renaming e iux eiux on the right-hand sides of and . . Convolution theorem. Main properties of the Fourier transforms. 1 . The convolution of two functions f x and g x is defined as 1 f f x g x f x - t g t dt. V2n J- By performing substitution x -1 u we see that the convolution is symmetric with respect to the convolved functions f x g x g x f x . The convolution theorem states that F f x g x F f x F g x . 2 . The main properties of the correspondence between functions and their Fourier transforms are gathered in Table . . Various Forms of the Fourier Transform 445 4. n-dimensional Fourier transform. The Fourier transform admits n-dimensional generalization 7 u 2n n 2 y f x e i u x dx u x U1X1 unxn where f x f x1 . xn f u f u1 . un and dx dx1 . dxn. The corresponding inversion formula is f x 2tt 2 7 u ei u x du du dui. dun. The Fourier transform is frequently used in the theory of linear partial differential equations with constant coefficients x e R . . Fourier Cosine and Sine Transforms 1. Fourier cosine transform. 1 . Let a function f x be integrable on the semiaxis 0 x oo. The Fourier cosine transform is defined by 2
