# complex values and residuee theorem

• May 9th 2011, 10:20 AM
eleahy
complex values and residuee theorem
I have two questions, I have tried to find stuff on them but failed
(1) find all complex values z such that e^(z+2)=1+i

I tried to put z=log(1+i)-2 but dont think thats not right

(2)use the residue theorem to evaluate
integralc(tanz)dz where c is the positively orientated circle |z|=2

i thought with the residue theorem you had to have a fraction so you could set the bottom to zero and get a singularity. I thought you could change tanz into sinz/cosz and set cosz to zero. but thats wrong

any help appreciated on either question
thanks :)
• May 9th 2011, 01:47 PM
Drexel28
Quote:

Originally Posted by eleahy
I have two questions, I have tried to find stuff on them but failed
(1) find all complex values z such that e^(z+2)=1+i

I tried to put z=log(1+i)-2 but dont think thats not right

It's right, assuming you realize that $\log(1+i)$ is a set, and not a number.

Quote:

(2)use the residue theorem to evaluate
integralc(tanz)dz where c is the positively orientated circle |z|=2

i thought with the residue theorem you had to have a fraction so you could set the bottom to zero and get a singularity. I thought you could change tanz into sinz/cosz and set cosz to zero. but thats wrong

any help appreciated on either question
thanks :)
Note that $\tan(z)$ has simple poles at $\pi\left(k+\frac{1}{2}\right)$ for all $k\in\mathbb{Z}$. Thus, find the number of them lying inside the disc $D(0,2)$ and use the residue theorem which says then that, if $p_1,\cdots,p_n$ are those finitely many poles then $\displaystyle \oint_{|z|=2}\tan(z)=2\pi\sum_j \text{Res}(\tan(z),p_j)$ and since each pole is simple you need only calculate the residue by $\text{Res}(\tan(z),p_j)=\lim_{z\to p_j}(z-p_j)\tan(z)$
• May 9th 2011, 02:31 PM
Mondreus
Calculating the residues can be done like this:
http://quicklatex.com/cache3/ql_c55a...d4dc37d_l3.png

To get the answer from the final part you can use the Maclaurin series for sin.