Entire functions
Definition. An entire function is holomorphic on all of \(\mathbb C\). Polynomials, exponentials, and trigonometric functions are entire; rational functions with finite poles are not.
Liouville theorem
Theorem. Every bounded entire function is constant. Cauchy's estimate gives \(|f'(a)|\le M/R\) on every circle of radius \(R\), and letting \(R\to\infty\) yields \(f'(a)=0\).
Fundamental theorem of algebra
Corollary. Every nonconstant complex polynomial has a root. If a polynomial \(p\) had no zeros, then \(1/p\) would be bounded entire, hence constant, forcing \(p\) constant.
Growth and degree
Proposition. If an entire function satisfies \(|f(z)|\le C(1+|z|)^m\), then \(f\) is a polynomial of degree at most \(m\). Cauchy's estimates force all derivatives of order greater than \(m\) to vanish.
Entire functions with omitted values
Remark. Picard's theorems refine Liouville dramatically: an entire non-polynomial function cannot omit too many complex values. This sits beyond the basic note but explains the strength of entire-function theory.
Examples
Example. The exponential function omits \(0\) but is unbounded. The sine function is entire and surjective onto \(\mathbb C\). Polynomials are exactly entire functions with a pole, rather than an essential singularity, at infinity.
Growth functions and order
For an entire function, define \[ M(r,f)=\max_{|z|=r}|f(z). \] The maximum principle makes \(M(r,f)\) nondecreasing. Its order is \[ \rho=\limsup_{r\to\infty} \frac{\log\log M(r,f)}{\log r}. \] Polynomials have order \(0\); \(e^{z^m}\) has order \(m\). Order measures global growth and controls the complexity of zero factorizations.
Fundamental theorem of algebra
Theorem. Every nonconstant complex polynomial has a zero.
Proof. If polynomial \(p\) had no zero, \(1/p\) would be entire. Since \(|p(z)|\to\infty\), \(1/p\to0\) at infinity and is bounded outside a large disc; it is bounded on the remaining compact disc. Liouville makes \(1/p\) constant, a contradiction.
Cauchy estimates and coefficient growth
If \(f(z)=\sum a_nz^n\), then \[ |a_n|\le\frac{M(r,f)}{r^n}. \] Optimizing over \(r\) translates growth bounds into coefficient decay. For \(M(r)\le Ce^{Ar^\rho}\), choosing \(r=(n/A\rho)^{1/\rho}\) gives a factorial-type bound on \(a_n\). Conversely, coefficient decay bounds reconstruct growth through the power series.
Jensen's formula
Theorem (Jensen). Suppose \(f\) is holomorphic near \(\overline{D(0,R)}\), \(f(0)\ne0\), and its zeros in the disc are \(a_k\), counted with multiplicity. Then \[ \log|f(0)| +\sum_{|a_k|<R}\log\frac R{|a_k|} =\frac1{2\pi}\int_0^{2\pi}\log|f(Re^{it})|\,\mathrm{d}t. \]
Proof. Divide \(f\) by the finite Blaschke-type factors that remove its zeros while preserving boundary modulus up to \(R/|a_k|\). The logarithm of the modulus of the resulting nonvanishing holomorphic function is harmonic. Apply its mean-value property at zero and rearrange.
Jensen's formula bounds the number of zeros in a smaller disc by growth on a larger circle. It is the elementary precursor of Nevanlinna theory.
Picard's global theorem
Little Picard states that a nonconstant entire function omits at most one complex value. Equivalently, an entire function omitting two values is constant. This sharpens Liouville, which requires the entire image to be bounded. A proof follows from the modular covering of \(\mathbb C\setminus\{0,1\}\) or from normal-family theory.