91Ó°ÊÓ

In Exercises 1-7, (a) plot the given function and find its Fourier transform. (b) If \(\hat{f}\) is real-valued, plot it; otherwise plot \(|\hat{f}|\). $$ f(x)= \begin{cases}2 x & \text { if } 0

Find the Fourier integral representation of the given function. $$ f(x)= \begin{cases}1 & \text { if } 0

In Exercises 7-20, solve the given problem. Assume that the functions in each problem have Fourier transforms. Take \(-\infty<0\). $$ \frac{\partial u}{\partial x}+3 \frac{\partial u}{\partial t}=0, \quad u(x, 0)=f(x) $$

In Exercises 7-12, find the Fourier sine transform of \(f(x)(x>0)\) and write \(f(x)\) as an inverse sine transform. Use a known Fourier transform and (10) when possible. $$ f(x)= \begin{cases}1 & \text { if } 0

In Exercises 7-12, find the Fourier sine transform of \(f(x)(x>0)\) and write \(f(x)\) as an inverse sine transform. Use a known Fourier transform and (10) when possible. $$ f(x)=x e^{-x^{2}} $$

In Exercises 1-12, use convolutions, the error function, and operational properties of the Fourier transform to solve the boundary value problem. Take \(-\infty0\). $$ \frac{\partial u}{\partial t}=-\frac{\partial^{4} u}{\partial x^{4}}, \quad u(x, 0)=f(x) $$

(a) Show that the rolution of the hent problem of Exarmple 1 with \(f(x)=T_{0}\) if \(0

Find the Fourier integral representation of the given function. $$ f(x)= \begin{cases}0 & \text { if }-1

In Exercises 7-20, solve the given problem. Assume that the functions in each problem have Fourier transforms. Take \(-\infty<0\). $$ a \frac{\partial u}{\partial x}+b \frac{\partial u}{\partial t}=0, \quad u(x, 0)=f(x) $$

Find the Fourier integral representation of the given function. $$ f(x)= \begin{cases}x & \text { if }-1