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.
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.
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.
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.
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.