Complex Analysis: Entire function dominated by another entire function

In summary, an entire function is a complex-valued function that is defined and analytic in the entire complex plane. It is possible for one entire function to be dominated by another entire function, meaning that the dominating function grows faster than the dominated function as the input increases. To determine if an entire function is dominated by another entire function, we can take the limit as the input approaches infinity and if the limit is 0, then the function is dominated. Examples of entire functions that are dominated by others include e^z dominated by z^2 and sin(z) dominated by e^z. This relationship can provide useful information about the behavior of the functions and simplify calculations.
  • #1
michael.wes
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Homework Statement



If f,g are entire functions and |f(z)| <= |g(z)| for all z, draw some conclusions about the relationship between f and g

Homework Equations



none

The Attempt at a Solution



I just need a push in the right direction.. thanks for any and all help!
 
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  • #2
Do you know Louiville's Theorem?
 
  • #3
Gib Z said:
Do you know Louiville's Theorem?

Thanks, I got it now :)
 

FAQ: Complex Analysis: Entire function dominated by another entire function

What is the definition of an entire function?

An entire function is a complex-valued function that is defined and analytic (has a derivative at all points) in the entire complex plane.

Can an entire function be dominated by another entire function?

Yes, it is possible for one entire function to be dominated by another entire function. This means that the dominating function grows faster than the dominated function as the absolute value of the input increases.

How can we determine if an entire function is dominated by another entire function?

To determine if an entire function f(z) is dominated by another entire function g(z), we can take the limit as the absolute value of z approaches infinity of f(z)/g(z). If this limit is equal to 0, then f(z) is dominated by g(z).

What are some examples of entire functions that are dominated by another entire function?

One example is the function f(z) = e^z, which is dominated by g(z) = z^2. Another example is f(z) = sin(z), which is dominated by g(z) = e^z. In both cases, the dominating function grows faster than the dominated function as the absolute value of z increases.

What is the significance of an entire function being dominated by another entire function?

An entire function being dominated by another entire function can provide useful information about the behavior of the functions. It can also be helpful in simplifying complex calculations and determining the convergence or divergence of certain series involving entire functions.

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