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<math>\hat{f}_c(w)=\sqrt{\frac{2}{\pi}}\left( | <math>\hat{f}_c(w)=\sqrt{\frac{2}{\pi}}\left( | ||
| − | \int_0^1(-1)\cos(wx)\, | + | \int_0^1(-1)\cos(wx)\,dx+ |
| − | \int_1^2(1)\cos(wx)\, | + | \int_1^2(1)\cos(wx)\,dx |
| − | \right) | + | \right)= |
</math> | </math> | ||
| + | |||
| + | <math>=\sqrt{\frac{2}{\pi}}\left([-\frac{1}{w}\sin(wx)]_0^1 | ||
| + | +[\frac{1}{w}\sin(wx)]_1^2\right)= | ||
| + | </math> | ||
| + | |||
| + | <math>=\sqrt{\frac{2}{\pi}}\ \frac{1}{w}\left( | ||
| + | -(\sin(w)-0)+(\sin(2w)-\sin(w)) | ||
| + | \right)=</math> | ||
| + | |||
| + | <math>=\sqrt{\frac{2}{\pi}}\ \frac{\sin(2w)-2\sin(w)}{w}.</math> | ||
| + | |||
| + | 517: 2. | ||
| + | |||
| + | <math>\hat{f}_c(w)=\sqrt{\frac{2}{\pi}}\left( | ||
| + | \int_0^k x\cos(wx)\,dx\right)= | ||
| + | </math> | ||
| + | |||
| + | <math>=\sqrt{\frac{2}{\pi}}\left(\left[\frac{x}{w}\sin(wx)+\frac{1}{w^2}\cos(wx)\right]_0^k | ||
| + | \right)= | ||
| + | </math> | ||
| + | |||
| + | <math>\sqrt{\frac{2}{\pi}}\left(\frac{k}{w}\sin(kw)+\frac{1}{w^2}\cos(kw) | ||
| + | -\frac{1}{w^2}\right).</math> | ||
| + | |||
| + | 517: 5. See page 2 of Bell's 11/10/2010 lecture at | ||
| + | [http://www.math.purdue.edu/~bell/MA527/Lectures/lec11-10.pdf Lesson 33] | ||
| + | |||
| + | 517: 7. See | ||
| + | [http://www.math.purdue.edu/~bell/MA527/HWK/p517_7.pdf p. 517: 7 Solution] | ||
| + | |||
| + | And for solutions to the three problems on p. 528, go to | ||
| + | [http://www.math.purdue.edu/~bell/MA527/jing Bell's Jing things] | ||
| + | |||
| + | [http://www.math.purdue.edu/~bell/MA527/prac2solns.pdf Exam 2 Practice Problem Solutions] | ||
Latest revision as of 12:06, 16 November 2010
Homework 12 Solutions
517: 1.
$ \hat{f}_c(w)=\sqrt{\frac{2}{\pi}}\left( \int_0^1(-1)\cos(wx)\,dx+ \int_1^2(1)\cos(wx)\,dx \right)= $
$ =\sqrt{\frac{2}{\pi}}\left([-\frac{1}{w}\sin(wx)]_0^1 +[\frac{1}{w}\sin(wx)]_1^2\right)= $
$ =\sqrt{\frac{2}{\pi}}\ \frac{1}{w}\left( -(\sin(w)-0)+(\sin(2w)-\sin(w)) \right)= $
$ =\sqrt{\frac{2}{\pi}}\ \frac{\sin(2w)-2\sin(w)}{w}. $
517: 2.
$ \hat{f}_c(w)=\sqrt{\frac{2}{\pi}}\left( \int_0^k x\cos(wx)\,dx\right)= $
$ =\sqrt{\frac{2}{\pi}}\left(\left[\frac{x}{w}\sin(wx)+\frac{1}{w^2}\cos(wx)\right]_0^k \right)= $
$ \sqrt{\frac{2}{\pi}}\left(\frac{k}{w}\sin(kw)+\frac{1}{w^2}\cos(kw) -\frac{1}{w^2}\right). $
517: 5. See page 2 of Bell's 11/10/2010 lecture at Lesson 33
517: 7. See p. 517: 7 Solution
And for solutions to the three problems on p. 528, go to Bell's Jing things
