Master Complex Analysis: Homework Statement, Equations, and Solutions

[Ted123]

Homework Statement

Homework Equations

The Attempt at a Solution

How do I go about Q1 and showing the coefficients are unique and then Q2?

 

[micromass]

What is your attempt at the solution?

 

[Ted123]

What is your attempt at the solution?

I was trying some sort of induction argument.

P(0)=Q(0) \implies a_0 = b_0 so the result is true for z=0

Now suppose for induction that P(k)=Q(k) ...

 

[micromass]

OK, so now take the derivative of the polynomials, and compare P^\prime(0) with Q^\prime(0).

 

[Ted123]

OK, so now take the derivative of the polynomials, and compare P^\prime(0) with Q^\prime(0).

Bearing in mind z is complex does normal differentiation still hold?

P'(0)=Q'(0) \implies a_1=b_1

 

[Ted123]

Scrap that method.

Could I say without loss of generality that n \geq m .

Then if P(z)=Q(z) then P(z)-Q(z)=0

So collecting up like terms we see that

P(z) - Q(z) = (a_0 - b_0) + (a_1 - b_1)z + (a_2 - b_2)z^2 + ... + (a_m - b_m)z^m + a_{m+1}z^{m+1} + ... + a_nz^n

This implies that a_j=b_j for all 0 \leq j \leq m (by equating coefficients).

So The first m terms are all 0.

So we have a_{m+1}z^{m+1} + ... + a_nz^n=0 .

Can you help me finish?

 

[Ted123]

On the other hand saying you can compare coefficients here might be a bit close to begging the question.

Instead, if P(z) = Q(z) for all z and F(z) = P(z)-Q(z), then 1, 2, 3, ..., m+n+1 are all roots of F(z), and so (z-1)(z-2)...(z-(m+n+1)) divides F(z) (factor theorem). Then what does this implies about the degree of F?