Math 421 Solution of Fall 99 Final

- Given that the first few terms of the Laurent series for the function
around are:

**a)**Find the principal part at of the function**Answer:**The part shown is the principal part.**b)**Find all the singularities of in the disk . Determine the nature of each singularity (isolated, removable, pole of what order, essential).**Answer:**Zero is a pole of order . The function can be written in the form , where Integral multiples of are simple zeroes (of order ) of the function , of which , , and are in the disk . The function does not vanish at and . Hence, and are simple poles of .**c)**Find the residue at each isolated singularity in .**Answer:**Using the Laurent serries above, we see that . At the simple pole , the residues is

- Compute
,
where is the circle
(traversed counterclockwise).
**Answer:**vanishes to order at integer multiples of (because ). The numerator has value at . Thus, has a simple pole at , . The only multiple of enclosed by is . Using the fact that is a simple pole, we get

Cauchy's Theorem yields,

- Compute
,
where is the circle
(traversed counterclockwise).
**Answer:**is an entire function, so its integral, over any closed contour, is zero (by Cauchy-Goursat's Theorem). Using the parametrization , we get

- Compute
**Answer:**Let . Then , and . The integral gets converted to the contour integral over the unit circle .

The integrand has poles at . Only is enclosed by . We get

- Compute
**Answer:**See the first Example in Section 60 of the text on page 205. **a)**Find the Laurent series of the function around the point .**Answer:**Let be the Taylor series of centered at . Taylor's Theorem states, in particular, that Now, and for . We get, for ,

. Summerizing, for we get

**b)**Find the Taylor series of the function around the point .**Answer:**Use the partial fraction decomposition

Now, use the Taylor series of to obtain

- Determine whether the following statements are true or false.
Justify your answers.
**a)**The limit exists and is equal to .**Answer:**False, is not analytic at . The above limit is the derivative limit, which does not exists. A direct argument, that the limit doesn't exists, consists of letting approach along the and axis:

**b)**There is a function , analytic in the disk , such that

**Answer:**False. Such a function would be a non-constant analytic function, whose absolute value achieves its maximum at the interior point of . This contradicts the*Maximum Modulus Principle*.**c)**If has an isolated singularity at and , then is a removeable singularity.**Answer:**False. Take and as a counter example. - Compute
Simplify your answer as much as
possible.
**Answer:**

- Prove that
where is the piece of the circle
going from
to counter-clockwise.
**Answer:**The curve is parametrized by , . Thus, has a positive imaginary part. Consequently, has a negative real part equal to , where , and

We conclude, that

- Find an entire function such that
.
**Answer:**Note the equality

Take .