Is this question incomplete? Regarding entire functions....

In summary, the conversation discusses a question about an entire function with given conditions, with hints to use Cauchy's Inequality and Maclaurin Series to show that the function may be a polynomial.
  • #1
Terrell
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Homework Statement


Let ##F## be an entire function such that ##\exists## positve constants ##c## and ##d## where ##\vert f(z)\vert \leq c+d\vert z\vert^n, \forall z\in \Bbb{C}##.

Is this question incomplete? My complex analysis course is not rigorous at all and this came up on a past final exam. Can anyone help me piece this question together? Or tell me what it could possibly be asking? Thanks.

Homework Equations


Hints: Use Cauchy's Inequality and Maclaurin Series of ##f##.

The Attempt at a Solution

 
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  • #2
Terrell said:

Homework Statement


Let ##F## be an entire function such that ##\exists## positve constants ##c## and ##d## where ##\vert f(z)\vert \leq c+d\vert z\vert^n, \forall z\in \Bbb{C}##.

Is this question incomplete? My complex analysis course is not rigorous at all and this came up on a past final exam. Can anyone help me piece this question together? Or tell me what it could possibly be asking? Thanks.

Homework Equations


Hints: Use Cauchy's Inequality and Maclaurin Series of ##f##.

The Attempt at a Solution


I'd guess they probably want you to show that ##f(z)## is a polynomial.
 
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Likes Terrell

1. Is it possible for an entire function to have a finite number of zeros?

Yes, it is possible for an entire function to have a finite number of zeros. One example of such a function is f(z) = ez - 1, which has only one zero at z = 0.

2. Are entire functions always analytic on the entire complex plane?

Yes, by definition, entire functions are analytic on the entire complex plane. This means that they are differentiable at every point in the complex plane.

3. Can an entire function have an essential singularity?

No, an entire function cannot have an essential singularity. This is because entire functions are defined to be analytic on the entire complex plane, and essential singularities occur when a function is not analytic at a certain point.

4. Are entire functions always bounded on the entire complex plane?

No, not all entire functions are bounded on the entire complex plane. For example, the function f(z) = ez grows without bound as z approaches infinity.

5. Is there a relationship between the zeros of an entire function and its growth rate?

Yes, there is a relationship between the zeros of an entire function and its growth rate. This is known as the Hadamard factorization theorem, which states that an entire function can be written as a product of its zeros and a function that grows at a certain rate.

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