Problem with Complex Polynomial

In summary, the conversation is about trying to solve an old homework problem which involves showing that an analytic function can be expressed as a polynomial in z. The person has tried various methods but has not been successful in obtaining the desired form. They are seeking help from others to solve the problem.
  • #1
xman
93
0
I am trying to figure out this old homework problem I haven't been able to solve. The problem goes like this:

Let [tex] f(z)=c_{00}+c_{10}x+c_{01}y+\cdots + c_{nm}x^{n}y^{m} [/tex] be a polynomial function of x and y. If, in addition, f is analytic function, show that f has to be a polynomial in z. Specifically, show that [tex] f(z)=a_0+a_1 z+\cdots +a_n z^n[/tex] where [tex] a_k=\frac{1}{k!} \frac{\partial^k f}{\partial x^k} (0)[/tex]

Solution: Now I know that if f is an analytic function in an open set, then f and its derivatives of order n are also analytic. Moreover, I know that if a function is n times differentiable and [tex] f^{(n)}(z)=0[/tex] for every z in some domain where f is analytic, then f in this domain is a polynomial of degree at most n-1.

Moreover, we also know that [tex] f^{\prime}(z) =f_{x}=-i f_y[/tex] where the subscripts on f represent derivatives with respect to that variable and f is assumed of the form [tex] f=u+i\,v[/tex]

My thoughts were, if I take say m+1 derivatives of f, then we satisfy [tex] f^{(m+1)}(z)=0[/tex] which guarantees we have a polynomial. Then, we we can rewrite by integrating each term pairwise, since we will have constants for each integration. But, this does not lead to an expression of the form we want to show.

So, I then tried writing f in terms of [tex] z, \overline{z}[/tex] but the cancellation of the [tex]\overline{z}[/tex] doesn't happen as I hoped and I again do not get the desired form.

I have tried a couple other algebra tricks, but fail to get the desired form. I was hoping someone could point me in the right direction to finally solve this problem.

Thanks in advance for any available help.
 
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  • #2
I can't say right off the top of my head, but I definitely noticed that [tex] f^{(m+1)}(z)=0[/tex] should be [tex] f^{(n+1)}(z)=0[/tex]

I'll think about it if nobody else comes up with an answer
 
  • #3
Office_Shredder said:
I can't say right off the top of my head, but I definitely noticed that [tex] f^{(m+1)}(z)=0[/tex] should be [tex] f^{(n+1)}(z)=0[/tex]

I'll think about it if nobody else comes up with an answer
Thanks for taking a look!

I don't believe it should matter, since m and n are just the order of the derivatives. So, I believe that in any three cases, i.e. m=n, m<n, and m>n, if f is analytic, and if the expressions of the higher order derivatives are of the form [tex] f^{(j)}(z)= f_{x^{(j)}} = (-1)^{j} f_{y^{(j)}}[/tex] then I believe if you can obtain derivatives with respect to either variable will eventually be zero, since derivatives of analytic functions are again analytic, then you can obtain the case where [tex] f^{(whatever)}(z)=0[/tex]
At least this is what I am assuming.
 

Related to Problem with Complex Polynomial

1. What is a complex polynomial?

A complex polynomial is a mathematical expression that contains one or more variables and coefficients, with the variables raised to positive integer powers. The coefficients and variables can take on both real and imaginary values, making it a complex number.

2. What is the problem with complex polynomials?

The main problem with complex polynomials is that they can have multiple solutions or roots. This means that there may be more than one value for the variable that satisfies the equation, making it difficult to find a single solution.

3. How do you solve a problem with complex polynomials?

To solve a problem with complex polynomials, you can use various techniques such as factoring, synthetic division, or the quadratic formula. It is important to remember that the solutions may be complex numbers and not just real numbers.

4. Why are complex polynomials important in science?

Complex polynomials are important in science because they can be used to model and solve a wide range of problems in various fields such as physics, engineering, and economics. They provide a more accurate representation of real-world phenomena that involve both real and imaginary quantities.

5. Can complex polynomials be graphed?

Yes, complex polynomials can be graphed on a complex plane, where the x-axis represents the real part of the complex number and the y-axis represents the imaginary part. The resulting graph is called a complex polynomial function, and it can have complex roots that are represented by the points where the graph intersects the x-axis.

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