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struggling with the following question:

Evaluate the following convolution integral

cos(x)*cos(x)

any help would be much appreciated, the hint of using the cos trig identity has been given.

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The hint of using trig identities is probably as follows $$\cos x*\cos x=\int_{\text{limits}^*}\cos (u')\cdot\cos(u-u')\,du'\\=\int_{\text{limits}}\cos u'(\cos u'\cos u+\sin u'\sin u)\,du'\\=\cos u\int_{\text{limits}}\cos^2u'\,du'+\sin u\int_{\text{limits}}\sin u'\cos u'\,du'$$ which can both be evaluated using identities relating to $\cos2u'$ and $\sin2u'$.

$*$ - I have written $\text{limits}$ since I'm not sure what regions you are defining $\cos u'$ on, so I suppose you can put this in.

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  • $\begingroup$ your first line should it be cos(u) not cos(u') $\endgroup$
    – George wall
    May 25, 2017 at 12:31
  • $\begingroup$ @Georgewall No, I am using $u'$ as the variable with respect to which I integrated (I did have $u$ and $u'$ the wrong way around in the other term, so understand the confusion. However it did not make any difference since $\cos$ is even) $\endgroup$
    – John Doe
    May 25, 2017 at 12:40

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