Math 421 Solution of Fall 99 Final

1. 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

2. 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,

3. 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

4. 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

5. Compute

Answer: See the first Example in Section 60 of the text on page 205.

6. 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

7. 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.

9. 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

10. Find an entire function such that .