Number of zero's of holomorphic function

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In summary, the conversation is about using Rouche's theorem and the argument principle to show that a holomorphic function with real values only on the real axis can have at most one zero in the open unit disc. The argument principle is used to count the possible zeros by considering the winding number of the contour around the boundary of the unit circle, and it is concluded that the only possible zeros are at +1 and -1.
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Homework Statement



Let f(z) be holomorphic in the open unit disc taking real values only on the real axis.show that f has at most one zero in the open disc.

Homework Equations



Rouche's theorem,the argument principle.

The Attempt at a Solution


obviously,f does not have non-real zero's .I need help in counting the zero's by using the argument principle.
 
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hedipaldi said:

Homework Statement



Let f(z) be holomorphic in the open unit disc taking real values only on the real axis.show that f has at most one zero in the open disc.

Homework Equations



Rouche's theorem,the argument principle.

The Attempt at a Solution


obviously,f does not have non-real zero's .I need help in counting the zero's by using the argument principle.

I'm not an expert, but is argument is zero at +1 and -1 and can't be zero anyplace else as the contour winds around the boundary of the unit circle. What are the possibilities for the winding number?
 

1. How can I determine the number of zero's of a holomorphic function?

The number of zero's of a holomorphic function can be determined by counting the number of distinct roots of the function, or by using the fundamental theorem of algebra which states that a polynomial of degree n has at most n distinct roots.

2. Can a holomorphic function have an infinite number of zero's?

No, a holomorphic function cannot have an infinite number of zero's. This is because a holomorphic function is a complex-valued function that is analytic in a region, and by the identity theorem, a holomorphic function is uniquely determined by its values in a neighborhood of a point. Therefore, a holomorphic function can only have a finite number of zero's within that region.

3. Can the number of zero's of a holomorphic function change?

Yes, the number of zero's of a holomorphic function can change under certain conditions. For example, if the function is allowed to vary over a larger region, the number of zero's may change. Additionally, if the function is allowed to take on complex values, the number of zero's may change as well.

4. Are there any techniques for finding the zero's of a holomorphic function?

Yes, there are several techniques for finding the zero's of a holomorphic function. One method is to use the Cauchy integral formula, which relates the values of the function to its zero's. Another method is to use the argument principle, which counts the number of zero's inside a given contour. Additionally, numerical methods such as Newton's method can be used to approximate the zero's of a holomorphic function.

5. Can a holomorphic function have non-real zero's?

Yes, a holomorphic function can have non-real zero's. This is because the concept of a zero applies to complex numbers as well, and a holomorphic function can have zero's at any complex number, including those with non-real components.

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