Infinite Product - Product Representations of Functions

Product Representations of Functions

One important result concerning infinite products is that every entire function f(z) (that is, every function that is holomorphic over the entire complex plane) can be factored into an infinite product of entire functions, each with at most a single root. In general, if f has a root of order m at the origin and has other complex roots at u1, u2, u3, ... (listed with multiplicities equal to their orders), then

f(z) = z^m \; e^{\phi(z)} \; \prod_{n=1}^{\infty} \left(1 - \frac{z}{u_n} \right) \; \exp \left\lbrace \frac{z}{u_n} + \frac12\left(\frac{z}{u_n}\right)^2 + \cdots + \frac1{\lambda_n}\left(\frac{z}{u_n}\right)^{\lambda_n} \right\rbrace


where λn are non-negative integers that can be chosen to make the product converge, and φ(z) is some uniquely determined analytic function (which means the term before the product will have no roots in the complex plane). The above factorization is not unique, since it depends on the choice of values for λn, and is not especially elegant. However, for most functions, there will be some minimum non-negative integer p such that λn = p gives a convergent product, called the canonical product representation. This p is called the rank of the canonical product. In the event that p = 0, this takes the form

f(z) = z^m \; e^{\phi(z)} \; \prod_{n=1}^{\infty} \left(1 - \frac{z}{u_n}\right).

This can be regarded as a generalization of the Fundamental Theorem of Algebra, since the product becomes finite and is constant for polynomials.

In addition to these examples, the following representations are of special note:

Sine function

Euler - Wallis' formula for π is a special case of this.

Gamma function

Schlömilch

Weierstrass sigma function

Here is the lattice without the origin.

Q-Pochhammer symbol

Widely used in q-analog theory. The Euler function is a special case.

Ramanujan theta function


An expression of the Jacobi triple product, also used in the expression of the Jacobi theta function

Riemann zeta function

Here pn denotes the sequence of prime numbers. This is a special case of the Euler product.

Note that the last of these is not a product representation of the same sort discussed above, as ζ is not entire.

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