第7讲数学变换与特殊函数的积分.ppt

数学实验七MATLAB中的数学变换与特殊函数的积分  Fourier变换与Fourier逆变换 Laplace变换与Laplace逆变换 Z变换与Z逆变换 特殊函数的积分 FOURIER(f,v) = F(v) = int(f(x)*exp(-i*v*x),x,-inf,inf). FOURIER(f,u,v) = F(v) = int(f(u)*exp(-i*v*u),u,-inf,inf). IFOURIER(F,u) = f(u) = 1/(2*pi) * int(F(w)*exp(i*w*u,w,-inf,inf) IFOURIER(F,v,u) = f(u) = 1/(2*pi) * int(F(v)*exp(i*v*u,v,-inf,inf), F = ZTRANS(f) ?F(z) = symsum(f(n)/z^n, n, 0, inf) F = ZTRANS(f,w) ?F(w) = symsum(f(n)/w^n, n, 0, inf). F = ZTRANS(f,k,w) ?F(w) = symsum(f(k)/w^k, k, 0, inf). 武汉理工大学数学系 尹强 一、Fourier变换和逆Fourier变换 syms t v w x fourier(1/t) returns i*pi*(Heaviside(-w)-Heaviside(w)) fourier(exp(-x^2),x,t) returns pi^(1/2)*exp(-1/4*t^2) fourier(exp(-t)*sym(Heaviside(t)),v) returns 1/(1+i*v) fourier(diff(sym(F(x))),x,w) returns i*w*fourier(F(x),x,w) syms t u w x ifourier(w*exp(-3*w)*sym(Heaviside(w))) returns 1/2/pi/(3-i*t)^2 ifourier(1/(1 + w^2),u) returns 1/2*exp(-u)*Heaviside(u)+1/2*exp(u)*Heaviside(-u) ifourier(v/(1 + w^2),v,u) returns i/(1+w^2)*Dirac(1,-u) ifourier(sym(fourier(f(x),x,w)),w,x) returns f(x) 二、Laplace变换和逆Laplace变换 syms a s t w x laplace(x^5) returns 120/s^6 laplace(exp(a*s)) returns 1/(t-a) laplace(sin(w*x),t) returns w/(t^2+w^2) laplace(cos(x*w),w,t) returns t/(t^2+x^2) laplace(x^sym(3/2),t) returns 3/4*pi^(1/2)/t^(5/2) syms s t w x y ilaplace(1/(s-1)) returns exp(t) ilaplace(1/(t^2+1)) returns sin(x) ilaplace(t^(-sym(5/2)),x) returns 4/3/pi^(1/2)*x^(3/2) ilaplace(y/(y^2 + w^2),y,x) returns cos(w*x) ilaplace(sym(laplace(F(x),x,s)),s,x) returns F(x) 三﹑Z变换与逆Z变换 syms k n w z ztrans(2^n) returns z/(z-2) ztrans(sin(k*n),w) returns sin(k)*w/(1-2*w*cos(k)+w^2) ztrans(cos(n*k),k,z) returns z*(-cos(n)+z)/(-2*z*cos(n)+z^2+1) ztrans(cos(n*k),n,w)