Definitions, Theorems, Examples, Topics for Asmar 2nd edition Chapter 7: 7.6 only 7.6 The Fourier Sine and Cosine transforms ==== u_xx + u_yy = 0 on |x|0, u(x,0)=f(x) The theory is in slides, http://www.math.utah.edu/~gustafso/s2013/3150/slides/ ch7fourierTransform.pdf EXAMPLE 1. Compute FCT[f], FST[f] then express f(x) by inverse transforms. f(x)=sin(x) pulse(x,0,Pi) ANSWER. FCT=sqrt(2/Pi)(cos(Pi w)+1)/(1-w^2) FST=sqrt(2/Pi) sin(Pi w)/(1-w^2) f(x)=sqrt(2/Pi) integral of FST*sin(wx), x=0 to x=inf f(x)=sqrt(2/Pi) integral of FCT*cos(wx), x=0 to x=inf EXAMPLE 2. Let f(x)=exp(ax), a>0, x>0. Find FCT[f] and express f by the inverse transform. ANSWER. Integrate by table. FCT=sqrt(2/Pi) a/(a^2+w^2). f(x)=(2a/Pi)integral of cos(wx)/(a^2+w^2), w=0 to w=inf EXAMPLE 3. Let f(x)=exp(-ax^2/2), a>0. Find FCT[f]. ANSWER. FCT=FT[exp(-ax^2/2)]=(1/sqrt(a))exp(-w^2/(2a)) by 7.2-theorem5. EXAMPLE 4. Compute FCT and FST of f(x)=a/(a^2+w^2) by (9) and (10). EXAMPLE 5. Find FST of f(x)=x exp(-x^2). EXAMPLE 6. Find FST of f(x)=x exp(-ax).