Tricky complex analysis questions....

[Zukias]

i.
Let f and g be functions with a pole at c. Create rules (and prove them) about how we can combine f and g at c.

and ii: Find the poles of the function :
$$\frac{cotz+cosz}{sin2z}$$

and classify these poles using part i.

 

[alyafey22]

Do you mean what happens to c when considering f+g ?

 

[Zukias]

Do you mean what happens to c when considering f+g ?

I think it means when functions are multiplied, added, substracted, divided etc, that's what I'm assuming anyway

 

[alyafey22]

I'll say write the Laurent expansion of f and g around the pole c and see what happens when you apply the arithmetic operations on the series.

 

[blue_raver22]

i. Rule 1: If f and g both have a pole of order m at c, then their sum f+g also has a pole of order m at c.

Proof: Let f(z) = \frac{a_m}{(z-c)^m} + ... + \frac{a_1}{z-c} + a_0 and g(z) = \frac{b_m}{(z-c)^m} + ... + \frac{b_1}{z-c} + b_0. Since both f and g have a pole of order m at c, we can write f(z) = \frac{a_m}{(z-c)^m} + h(z) and g(z) = \frac{b_m}{(z-c)^m} + k(z), where h and k are analytic at c. Then, f+g = \frac{a_m+b_m}{(z-c)^m} + h(z) + k(z), which has a pole of order m at c.

Rule 2: If f has a pole of order m at c and g has a pole of order n at c, where m>n, then their product f*g has a pole of order m at c.

Proof: Let f(z) = \frac{a_m}{(z-c)^m} + ... + \frac{a_1}{z-c} + a_0 and g(z) = \frac{b_n}{(z-c)^n} + ... + \frac{b_1}{z-c} + b_0. Since m>n, we can write f(z) = \frac{a_m}{(z-c)^m} + h(z) and g(z) = \frac{b_n}{(z-c)^n} + k(z), where h and k are analytic at c. Then, f*g = \frac{a_mb_n}{(z-c)^{m+n}} + ... + \frac{a_1b_n}{(z-c)^{n+1}} + \frac{a_mb_1}{(z-c)^{m+1}} + h(z)k(z), which has a pole of order m at c.

ii. The function \frac{cotz+cosz}{sin2z} can be rewritten as \frac{cosz}{sinzsin2z} = \frac{1}{2