Thread: 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

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 is a set, and not a number.

(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 has simple poles at for all . Thus, find the number of them lying inside the disc and use the residue theorem which says then that, if are those finitely many poles then and since each pole is simple you need only calculate the residue by

Calculating the residues can be done like this:


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