Deriving the integration formula int^pii_0 e^(a cos(t)) * cos(a sin (t)) dt = pi?

I have this formula in my text;





int_c e^(az) / z dz = 2*pi*i (for any real constant a) and the C is a unit circle z=e^(i*t) where - pi %26lt;= t %26lt;= pi





which I had to prove as part of the first part of the question, now it asks me to re-write that in terms of theta to derive the "integration" formula which it gives as





int^pii_0 e^(a cos(t)) * cos(a sin (t)) dt = pi





Now for me, writing something in terms of pi just means to put pi where the z went...





But if I did that i end up with





int_c e^(a t) / t dt = 2*pi*i





how do I put sin's and cos's into the formula when there is no e^(i*t) ?? Any suggestions... :-)

Deriving the integration formula int^pii_0 e^(a cos(t)) * cos(a sin (t)) dt = pi?
You need to make use of the following relations:





on the unit circle, z = e^(i*theta) = cos(theta) + i sin(theta)





write the z in terms of exponents of theta.





Then separate the real and imaginary terms.





e^az = e^[acos t + iasint]


z = e^(it)





So (e^az)/z = e^[acos t + i(asint - t)]


= e^acost * e^[i(asint - t)]


= e^acos(t) * { cos [asint - t] + i sin [asint - t] }





Try rearranging this term and combine it with the integral you just proved.

gardenia