Question

Seien \(f\) und \(g\) elliptische Funktionen zum selben Gitter. a) Wenn \(f\) und \(g\) dieselben Pole und dieselben Hauptteile in den Polen haben, so unterscheiden sie sich nur um eine additive Konstante. b) Haben \(f\) und \(g\) dieselben Pol- und Nullstellen jeweils mit denselben Vielfachheiten, so unterscheiden sie sich um eine multiplikative Konstante.

Short answer

If their poles and principal parts match, \(f\) and \(g\) differ by an additive constant. If poles and zeros match in multiplicities, they differ by a multiplicative constant.

Step-by-step solution

Understanding the Problem

We need to show two statements about elliptic functions \(f\) and \(g\) related to their poles and zeros. Each function is defined over the same lattice.

Statement a: Identifying Conditions

We are told that the functions \(f\) and \(g\) have the same poles with identical principal parts at those poles. This implies, algebraically, that the difference \(h = f - g\) would be holomorphic everywhere, since any pole of \(f\) is also a pole of \(g\) with the same principal part.

Statement a: Concluding with Additivity

Since the function \(h\) is holomorphic everywhere on the torus and both \(f\) and \(g\) are periodic, \(h\) must also be periodic and bounded, which implies by Liouville's theorem that \(h\) is constant. Thus, \(f\) and \(g\) differ by an additive constant.

Statement b: Identifying Conditions

Here, \(f\) and \(g\) are specified to have the same poles and zeros (with the same multiplicities). This implies the quotient \(h = \frac{f}{g}\) is holomorphic and non-zero everywhere, because the poles of \(f\) match exactly with the zeros of \(g\) and vice versa.

Statement b: Concluding with Multiplicativity

Since \(h\) is holomorphic everywhere, it is also periodic and bounded. By Liouville's theorem, \(h\) must be a constant. Thus, \(f\) is a constant multiple of \(g\).

Key concepts

Principal Part

In complex analysis, the "Principal Part" of a function plays a crucial role when dealing with poles. When a function has a pole at a point, it means the function is not defined at that point but can be expressed as a Laurent series. The principal part consists of the terms with negative powers in this series. This part captures the behavior of the function around the pole.

Understanding the principal part is important because if two elliptic functions have the same principal parts at their poles, we can infer essential properties about their differences. Specifically, as in our problem, if two functions share identical principal parts at every common pole, their difference is a bounded function, which is also holomorphic. This boundedness then leads us to apply Liouville's theorem.

Liouville's Theorem

Liouville's Theorem is a powerful tool in complex analysis. It states that any bounded holomorphic function on the entire complex plane must be constant. This theorem helps simplify problems by restricting the nature of functions that meet specific criteria.

In the context of elliptic functions, we use this theorem when the difference between two functions, defined on the same lattice, is found to be bounded and holomorphic. With these conditions, Liouville's theorem tells us that the function must be constant across the entire domain. Therefore, if two elliptic functions differ only by a bounded holomorphic function, they differ by a constant—a vital step in solving part (a) of our exercise.

Holomorphic Functions

A holomorphic function is a complex-valued function that is differentiable at every point in its domain. This property makes holomorphic functions extremely smooth and predictable in their behavior.

When working with elliptic functions, holomorphicity is a key property that assures us of the function's continuity and differentiability within domains. In our problem, we determine that the difference of two elliptic functions with the same principal parts at their poles is holomorphic. This holomorphic nature allows us to apply Liouville's theorem effectively.

• They are smooth, having derivatives of all orders.
• They abide by the property of containing no oscillations or jumps.
• They're predictable when derived, with Laurent series expansions around singularities.

Lattice Theory

Lattice Theory provides the framework for understanding the periodicity of elliptic functions. A lattice in the complex plane is a discrete set of points with a repeating, grid-like structure formed by two basis vectors, making it possible for functions like the Weierstrass function or elliptic functions to be doubly periodic.

In our exercise, elliptic functions are defined over the same lattice. This commonality implies that their periodicity patterns and other lattice-dependent properties, such as pole positions, can be compared directly. This aspect of lattice theory assures us that periodicity conditions, necessary for applying other theorems, like Liouville's, are consistent for the functions in question, leading us to conclusions about their differences.