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Entire Function

Entire functions are those functions which are related to the field of complex analysis. Any entire function is also known as an integral function and hence both these terms are synonyms. In simple words, an entire function is defined as a function of complex values, which is holomorphic (differentiable at almost each and every point of its domain) on the complete complex plane. The most basic example of an entire function is the exponential functions, as they are holomorphic over the entire complex plane.

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Definition

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As we know by the definition of a function f is defined as a rule which gives only one value to every element of a given set A to a given set B, and to define an entire function we need to know about the complex valued functions which are made up of complex variables, in which a complex variable is defined as follows:
z = x + iy, where the real part is represented by the value of x and the imaginary part (which is also real valued) is represented by y as it is linked with iota, i.

Now we can define a complex valued function by denoting the value of the function f at any given point z as equal to w, which can be mathematically written as w = f (z), and hence here f (z) denotes a complex valued function.
Before defining an entire function we also need to know about the analyticity of a complex valued function, which is defined as follows:

Any complex valued function f (z) defined on an open set G, is said to be analytic if this function f (z) has a derivative at each and every point of G.
Thus, we can always say that any complex valued function will be considered analytic only if the function is defined over some open sets, where differentiability can hold true at only one single point only. Hence, when we say that a function f (z) is analytic at the point z0, it implies that the function f (z) is differentiable in some neighborhood of the point z0.

Now we can define the entire functions by applying the above mentioned definitions, in the form of the following definition:
A complex valued function f (z) is said to be an entire function if and only if the function f (z) is analytic on the complete complex plane.The mathematical definition of the entire function is defined as follows:
A function f (z) is said to be an entire function if it has a representation of the kind of the following:
f (z) = Sigma varying from k = 0 till infinity, ak * zk, where ak = f k (0) / k!, k $\geq$ 0, which is valid for $\left | z \right |<\infty $.
These functions are also represented by E, which are known as linear space.
We can also note that if f (z) is not equal to zero anywhere, then the function f (z) is always equal to the value of eg(z), where g (z) is another function representing an entire function. And similarly, we can generalize the above definition as, if there are finitely many points at which the function f (z) takes the value of zero, then all these points, which are also known as zeroes of the function f (z), are represented by z1, z2, ….,zk, and then,

f (z) = (z – z1) * (z – z2) ….(z - zk) * eg(z), where g (z) is an entire function.
We can also find out that a function consisting of many variables, represented by f ( z2, z2, …, zn ) will be an entire function if and only if this function of many variables is analytic for | zk | < infinity, and k = 1, 2, …., n.

Let’s consider another function f (z) = Sigma varying from k = 0 till infinity, ak * zk, then this function will be entire if and only if it satisfies the following:
Limit n tending to infinity, $\left | a_{n} \right |^{\frac{1}{n}} $ = 0. Thus, the complex valued function f (z) becomes an entire function if it satisfies the above mentioned criteria.

Properties

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The most basic property of an entire function which comes from the very basic definition of the entire function is that each and every entire function can be represented as a power series in the variable z, which is convergent everywhere.

This basic property gives rise to the following important properties of any given entire function:
Property Number 1:
If a complex valued function f is an entire function satisfying the following criteria:

$g (z)$ = $\frac{f (z) – f (a)}{ (z – a}$, when z is not equal to a, and
$g (z)$ = $f ‘ (a)$, when z is equal to a, then

the integral over C, of g(z) dz = 0, where C is the boundary of any rectangle R.
Also, here g (z) is the derivative of some entire function, which is also known as the integral theorem and the integral which is around any closed curve vanishes as it approaches to g (z), which is also know as the closed curve theorem.

Property Number 2:  Let the function f is an entire function and a is any given complex number and C is the curve defined as C: Reiq, 0 < = x < = 2pi, with R > $\left | a \right |$ such that a is always in the interior of C, then,
f (a) = {1 / i* 2pi} integral over C of [f (z) / (z -a)] dz.

Note that this is the most important property in the filed of complex analysis as it always gives an approach to integrate those complex valued functions which are not entire functions, just by applying the above stated Cauchy integral formula.
As a result of the above property, we can find that if a is any point that is contained in the circle C with the centre at g, and radius r, then
the integral over C, of [1 / z - a] dz = i * 2 pi.

Property Number 3: This property is known as the Taylor expansion of any given entire function. With the help of the above property number 2, that is, by using the Cauchy’s formula, we can find that any given entire complex function can be represented y a power series expansion. Thus, the statement of property number 3 is as following:
 Let the complex valued function f (z) is an entire function, then it will have a power series representation and also f k (0) exists for every point k, and
f (z) = Sigma varying from k = 0 till infinity, [f ^ k (0) / k !] * z^k.
As a result of the above property we can conclude that any entire function is infinitely differentiable and also if f is an entire function and a is any given complex number, then the function f (z) = f (a) + f' (a) ( z – a ) + [f'' (a) / 2!] * ( z – a )2 + …… for all z.

Property Number 4: Any given bounded entire function is always constant which is also known as the famous Liouvulle’s Theorem.

Property Number 5: The sums and the products of two or more than two entire functions are also an entire function. Lets consider the function f (z) = z, which is an entire function then we can find that any polynomial of the form P(z) = sigma from n varying from 0 to N of [an * zn] of finite degree N, will also be an entire function.

Proof

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In order to show any proof of an entire function we need to know certain notations along with some theorems and principles. These are mentioned below:
Some of the Notations that are used are: As we know that the open Euclidean disc is represented by:
        D (z0, r) = {z $\in$ C: | z – z0 | < r, for z0 $\in$ C and r > 0.
Similarly, the circle of the center z0, with radius r, is represented by
        S (z0, r) = {z $\in$ C: | z – z0 | = r.
If z0 $\in$ C* = C $\cup$ $\infty$, then:
         D (z0, r) = {z $\in$ C*: q ( z, z0 ) < r, will represent the spherical disc.

The principle of maximum modulus: Whenever a function f is an entire function, where we define M (r; f) = max { | f(z) | : | z | = r }, then by applying the principle of maximum modulus, we will find that M (r; f) = max { | f(z) | : | z | $\leq$ r }, thus making the function M (r, f) a non decreasing function.

Theory of Wiman – Valiron: This theory is very important for proving almost all kinds of problems relating to the complex valued entire functions as this theorem finds out the local behavior of the given entire function from relating it to its power series.
Consider the polynomial function of positive degree n, as P (z) = an * zn + … + a0, where an is not equal to zero, and if z and z0 are large, then:
P (z) is represented by (z / z0) $^n$ * P (z0) and
P’ (z) / P (z) represents n / z.
Thus, if P is replaced by an entire function f, which is non polynomial, then by applying the Picard’s theorem, we will find that for large z and z0, there exists no such type of asymptotic rule, but on the other hand, the purpose of the Wiman Valiron theory is to find such values when z is near to z0 and | f (z0) | is close to M (| z0 |, f).
Considering f (z) = sigma varying from k from 0 to infinity of (ak * z$^k$) as a transcendental entire function, which means a non rational function, then ak is not equal to zero for infinitely many k.

Formula of Cauchy Estimate: By applying this formula, as discussed in the above properties of entire functions, we can find a result that whenever a function f is analytic in the complete complex plane, that is, whenever the function f is an entire function and which is bounded too, such that it satisfies,
 |f (z)| $\leq$ M, for all z, then by applying the above Cauchy estimate formula, at any point of z0 in the complex plane, we can find that its derivative will always be bounded by the following:
|f ‘( z0 )| $\leq$ M / R.
As the function is an entire function, we can take any possible large value of R. Therefore, we got that | f’ (z0) | = 0, whenever f’ (z0) = 0, and since this holds true for all z0 in the whole complex plane, thus as a result, we have proven the statement of another important theorem know as Liouville’s Theorem.
Thus, all these principles and theorems are must for the basic understanding of proving the entire functions.

Entire Function Problems

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The following are some of the problems based on entire functions which are explained below:

Problem 1: The residue of any given entire function at infinity is always zero.

Solution: The above statement holds true for all entire functions, because of the very definition of the residue at infinity along with the theorem based on the Cauchy – Goursat Theorem. An alternative way to solve this problem is to expand the function using Taylor expansion for the function f at the point zero.

f ( z ) = $\sigma$ from n $\in$ [0, $\infty$) of [an * zn] and then replacing z with 1/z in order to get the Laurent expansion of the function f (1/z) at the point 0, which will give f ( z ) = sigma from n $\in$ ( -$\infty$, 0] of [a(-n) * zn.
Thus, as a result we will get the residue at infinity which will be equal to the negative of the coefficient of 1/z, term of the following:

(1 / z2) * f ( 1/z ) = sigma from n  $\in$ ( -$\infty$, -2] of [a(-n-2) * zn, which is obviously equal to zero.

Problem 2: Let f and g are two given entire functions such that | f (z) | $\leq$ | g (z) | for all z belonging to C. Then show that the function f = cg, for some value of c belonging to C.

Solution: Consider if the function g (z) = 0, for all z belonging to C, then the above result is obviously true. On the other hand, the zeroes of the function g are isolated and the quotient function defined as h (z) = f (z) / g (z), which is bounded in each of the deleted neighborhood of every zeroes of the function g.

Thus, by applying the Riemann’s Principle, the function h extends uniquely to an entire function, and thus, by the definition of continuity, we find the following:

| h (z)| $\leq$ 1 for all values of z belonging to C, and thus by applying the Liouville’s theorem, the function h is constant, and therefore,

f ( z ) = c * g (z) for all the values of z belonging to C.
Hence, the above result also holds true when g (z) is not equal to zero too.

Problem 3: If u: R2 $\rightarrow $ R is given to be a positive harmonic function, then show that u is always a constant.

Solution: Lets consider v to be a harmonic conjugate of u on R2 = C. Then we will find that the function f = u + iv will be an entire function, where u $\geq$ 0.
Now by applying Liouville’s theorem on the function e-f, that the function f is a constant function. Hence, the above result always hold true.

Problem 4: Show that the function exp (z) = exp (x + i y) = ex (cos y + i sin y) is an entire function.

Solution: The first thing is to check whether the above function satisfies the Cauchy – Riemann equations or not. Thus, we find that del (u) / del (x) = ex * cos y = del (v) / del (y) and del (v) / del (x) = ex * sin y = - del (u) / del (y).
Hence, Cauchy – Riemann equations are satisfied everywhere on the whole complex plane. Therefore, the function is an entire function, and its derivative will be found as:
exp' (z) = del (u) / del (x) + i [del (v) / del (x)],
    = ex * cos y + i ex * sin y = exp (z).

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