Complex Analysis Entire Functions

Thread starter bballife1508
Start date Dec 14, 2010
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Analysis Complex Complex analysis Functions

In summary, a complex analysis entire function is a function that is analytic on the entire complex plane, meaning it is differentiable at every point. These functions are important in mathematics due to their useful properties and applications in various fields. To determine if a function is entire, one can check if it is differentiable at every point or if it has a convergent power series representation. While every complex polynomial is an entire function, not every entire function is a complex polynomial due to the potential for infinitely many terms in the power series representation. Lastly, entire functions cannot have poles or essential singularities because they are defined and differentiable at every point in the complex plane.

Dec 14, 2010

bballife1508

Let f(z) be an entire function such that |f(z)| less that or equal to R whenever R>0 and |z|=R.

(a)Show that f''(0)=0=f'''(0)=f''''(0)=...

(b)Show that f(0)=0.

(c) Give two example of such a function f.

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Dec 14, 2010

snipez90

Hint: look back through your old post history. I already explained one tool you could use to approach these problems.

1. What is a complex analysis entire function?

A complex analysis entire function is a function that is analytic (can be represented by a convergent power series) on the entire complex plane. This means that the function is differentiable at every point in the complex plane.

2. What is the significance of entire functions in mathematics?

Entire functions are important in mathematics because they have many useful properties, such as being infinitely differentiable and having a Taylor series representation. They also have applications in fields such as physics and engineering.

3. How do you determine if a function is entire?

A function can be determined to be entire by checking if it is differentiable at every point in the complex plane. If it is differentiable at every point, then it is an entire function. Additionally, if a function can be represented by a convergent power series, it is also an entire function.

4. What is the relationship between entire functions and complex polynomials?

Every complex polynomial is an entire function, but not every entire function is a complex polynomial. This is because an entire function can have infinitely many terms in its power series representation, while a polynomial only has a finite number of terms.

5. Can entire functions have poles or essential singularities?

No, entire functions cannot have poles or essential singularities. This is because they are defined and differentiable at every point in the complex plane, so there are no points where the function is undefined or non-differentiable.