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 \(|\hat{f}|\). $$ f(x)= \begin{cases}1-|x| & \text { if }|x| \leq 1 \\ 0 & \text { otherwise }\end{cases} $$

In Exercises \(1-6\), determine the solution of the given wave or heat problem. Give your answer in the form of an inverse Fourier transform. Take the variables in the ranges \(-\infty<0\). $$ \frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}}, \quad u(x, 0)=\sqrt{\frac{2}{\pi}} \frac{\sin x}{x}, \quad \frac{\partial u}{\partial t}(x, 0)=0 $$

Find the Fourier integral representation of the given function. $$ f(x)=e^{-|x|} $$

In Exercises 1-6, find the Fourier cosine transform of \(f(x)(x>0)\) and write \(f(x)\) as an inverse cosine transform. Use a known Fourier transform and (9) when possible. $$ f(x)= \begin{cases}\cos x & \text { if } 0

In Exercises 5-12, plot the given function and find its derivative. Use Dirac deltas where appropriate. Take \(n=0,1,2, \ldots, a\) and \(\alpha\) real numbers with \(a>0\). $$ \phi(x)= \begin{cases}1 & \text { if } x<-1 \\ 0 & \text { if }-11\end{cases} $$

In Exercises 5-12, plot the given function and find its derivative. Use Dirac deltas where appropriate. Take \(n=0,1,2, \ldots, a\) and \(\alpha\) real numbers with \(a>0\). $$ \phi(x)= \begin{cases}1 & \text { if } x<-1 \\ x^{2} & \text { If }-11\end{cases} $$

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 \(|\hat{f}|\). $$ f(x)= \begin{cases}1-\frac{|x|}{a} & \text { if }|x| \leq a \\ 0 & \text { otherwise }\end{cases} $$ where \(a>0\).

In Exercises 1-6, find the Fourier cosine transform of \(f(x)(x>0)\) and write \(f(x)\) as an inverse cosine transform. Use a known Fourier transform and (9) when possible. $$ f(x)= \begin{cases}1-x & \text { if } 0

Find the Fourier integral representation of the given function. $$ f(x)= \begin{cases}1-x^{2} & \text { if }-1

In Exercises \(5-14\), solve the given problem. $$ \begin{gathered} u_{t}=u_{x x}+x e^{-t} \\ u(x, 0)=0 \end{gathered} $$