Calculating Line Integral on Unit Circle with Complex Analysis Method

[Doonami]

Good old complex analysis. I'm trying to evaluate a line integral which looks like this

$$\oint$$e ^{(z + [1$$/$$z])} for |z| = 1

So I guess I'm dealing with a circle with a radius 1, so I've parameterised:

z = e^it

I need to sub this into my formula of:

$$\int$$_c f(z)dz = $$\int$$f(z(t)) z'(t)dt

(this is from [0,2pi]

However, when I go to sub that in I get an integral of an exponential to the power of an exponential. Can anyone suggest how to do that?

 

[tiny-tim]

Welcome to PF!

Good old complex analysis. I'm trying to evaluate a line integral which looks like this

$$\oint$$e ^{(z + [1$$/$$z])} for |z| = 1

So I guess I'm dealing with a circle with a radius 1, so I've parameterised:

z = e^it

I need to sub this into my formula of:

$$\int$$_c f(z)dz = $$\int$$f(z(t)) z'(t)dt

(this is from [0,2pi]

However, when I go to sub that in I get an integral of an exponential to the power of an exponential. Can anyone suggest how to do that?

Hi Doonami ! Welcome to PF!

Hint: go for the obvious … substitute u = 1/z (and be very careful about the limits of integration).

And cryptic hint: Then compare it with the derivative of the integral.

 

[HallsofIvy]

Yes. z= e^it so e^{z+ 1/z} becomes
$$e^{z+ 1/z}= e^{e^{it}+ e^{-it}}= e^{\frac{e^{2it}+ 1}{e^{it}}$$
If you let u= e^it then du= ie^itdt so -idu/u= dt.