Complex analysis - something really confusing

[sweetvirgogirl]

I think I have misunderstood one of the theorems in complex analysis

(k reperesents the order of the derivative)


Theorem: Suppose f is analytic on a domain D and, further, at some point z0 subset of D, f (k) (z0) = 0. Then f(z) = 0 for all z subset of D ...

Is the theorem basically saying is that if f(z) equals 0 at any z0, then it will equal zero for all of the points?? That doesnt' sound right at all ...

any help with be greatly appreciated

 

[leach]

That theorem is obviously wrong. The function [tex]f(z) = z^{k+1}[/tex] is a counterexample. It verifies [tex]f^{k}(0) = 0[/tex], is holomorphic in any domain, yet it is nonzero for every [tex]z\ne 0[/tex].

Most probably your theorem is one of the following:

1) If [tex]f^k(z_0) = 0[/tex] for all [tex]k\ge 0[/tex], then f=0 in D.

2) If [tex]f=0[/tex] in some open subset of D, then f=0 in D.

 

[tacman]

It's understandable that this theorem may seem confusing at first glance. However, let's break it down step by step. First, we need to understand what it means for a function to be analytic on a domain D. This means that the function is differentiable at every point in that domain.

Next, the theorem states that if the kth derivative of the function f is equal to 0 at some point z0 in the domain D, then the function f must be identically equal to 0 for all points in that domain.

To understand this better, let's consider an example. Suppose we have a function f(z) = z^2 + 2z + 1, which is analytic on the complex plane. This means that it is differentiable at every point on the complex plane. Now, let's take the second derivative of this function, f''(z) = 2. If we evaluate this at any point, say z = 1, we get f''(1) = 2. This means that the second derivative of the function is equal to 0 at z = 1.

According to the theorem, this means that the function f(z) must be identically equal to 0 for all points in the complex plane. However, we know that this is not true because f(1) = 4, which is not equal to 0.

So, where did we go wrong? We misunderstood the theorem. The theorem only applies when the kth derivative of the function is equal to 0 at a specific point. In our example, we took the second derivative of f and evaluated it at z = 1, which is not the same as saying that f''(z) = 0 for all points in the complex plane.

In summary, the theorem is saying that if the kth derivative of a function is equal to 0 at a specific point, then the function must be identically equal to 0 for all points in the domain. It does not apply to all points in the domain, only at the specific point where the kth derivative is equal to 0.

I hope this explanation helps clear up any confusion. Remember, complex analysis can be complex (no pun intended), but with practice and patience, you'll be able to understand and apply these theorems accurately.