數學七matlab中的數學變換與特殊函數的積分

1、2000-2001,數學實驗七MATLAB中的數學變換與特殊函數的積分,Fourier變換與Fourier逆變換 Laplace變換與Laplace逆變換 Z變換與Z逆變換 特殊函數的積分,,一、Fourier變換和逆Fourier變換,2000-2001,FOURIER(f,v) F(v) = int(f(x)*exp(-i*v*x),x,-inf,inf).FOURIER(f,u,v) F(v) = int

2、(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),,syms t v w xfourier(1/t) returns i*pi*(Heavisi

3、de(-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)

4、returns i*w*fourier(F(x),x,w),syms t u w xifourier(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)

5、 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變換和逆Laplace變換,syms a s t w x laplace(x^5) ret

6、urns 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

7、 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*

8、x)ilaplace(sym('laplace(F(x),x,s)'),s,x) returns F(x),,三﹑Z變換與逆Z變換,2000-2001,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

9、 = ZTRANS(f,k,w) ?F(w) = symsum(f(k)/w^k, k, 0, inf).,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) re

10、turns z*(-cos(n)+z)/(-2*z*cos(n)+z^2+1) ztrans(cos(n*k),n,w) returns w*(-cos(k)+w)/(-2*w*cos(k)+w^2+1) ztrans(sym('f(n+1)')) returns z*ztrans(f(n),n,z)-f(0)*z,iztrans(z/(z-2)) returns 2^n

11、 iztrans(exp(x/z),z,k) returns x^k/k!,,文本格式轉換 Latex(f) Ccode(f) Fortran(f),syms x f = taylor(log(1+x)); latex(f) = x-1/2\,{x}^{2}+1/3\,{x}^{3}- 1/4\,{x}^{4}+1/5\,{x}^{5}

12、 H = sym(hilb(3)); latex(H) = \left [\begin {array}{ccc} 1&1/2&1/3 \\\noalign{\medskip}1/2&1/3&1/4 \\\noalign{\medskip}1/3&1/4&1/5\end

13、 {array}\right ],syms x f = taylor(log(1+x)); ccode(f) = t0 = x-x*x/2+x*x*x/3-pow(x,4.0)/4+pow(x,5.0)/5; H = sym(hilb(3)); ccode(H) = H[0][0] = 1.0; H[0][1] = 1.0/

14、2.0; H[0][2] = 1.0/3.0; H[1][0] = 1.0/2.0; H[1][1] = 1.0/3.0; H[1][2] = 1.0/4.0; H[2][0] = 1.0/3.0; H[2][1] = 1.0/4.0; H[2][2] = 1.0/5.0;,syms x f = taylor(log(1+x)); fortran(f)

15、 = t0 = x-x**2/2+x**3/3-x**4/4+x**5/5 H = sym(hilb(3)); fortran(H) = H(1,1) = 1 H(1,2) = 1.E0/2.E0 H(1,3) = 1.E0/3.E0 H(2,1) = 1.E0/2.E0 H(2,2) = 1.E0/3.E0 H(2,3) = 1.E0/4.E0 H(3