A function of a complex variable is holomorphic in an open domain if it is differentiablewith respect to at every point of. Equivalently, it is holomorphic if it is analytic, that is, if its Taylor series exists at every point of, and converges to the function in some neighbourhood of the point. A function is meromorphic in if every point of has a neighbourhood such that either or is holomorphic in it. A zero of a meromorphic function is a complex number such that. A pole of is a zero of. If is a function that is meromorphic in a neighbourhood of a point of the complex plane, then there exists an integer such that is holomorphic and nonzero in a neighbourhood of . If, then is a pole of order of. If, then is a zero of order of. Simple zero and simple pole are terms used for zeroes and poles of order Degree is sometimes used synonymously to order. This characterization of zeros and poles implies that zeros and poles are isolated, that is, every zero or pole has a neighbourhood that does not contain any other zero and pole. Because of the order of zeros and poles being defined as a non-negative number and the symmetry between them, it is often useful to consider a pole of order as a zero of order and a zero of order as a pole of order. In this case a point that is neither a pole nor a zero is viewed as a pole of order 0. A meromorphic function may have infinitely many zeros and poles. This is the case for the gamma function, which is meromorphic in the whole complex plane, and has a simple pole at every non-positive integer. The Riemann zeta function is also meromorphic in the whole complex plane, with a single pole of order 1 at. Its zeros in the left halfplane are all the negative even integers, and the Riemann hypothesis is the conjecture that all other zeros are along. In a neighbourhood of a point a nonzero meromorphic function is the sum of a Laurent series with at most finite principal part : where is an integer, and Again, if , one has a pole of order, and if , one has a zero of order.
At infinity
A function is meromorphic at infinity if it is meromorphic in some neighbourhood of infinity, and there is an integer such that exists and is a nonzero complex number. In this case, the point at infinity is a pole of order if, and a zero of order if. For example, a polynomial of degree has a pole of degree at infinity. The complex plane extended by a point at infinity is called the Riemann sphere. If is a function that is meromorphic on the whole Riemann sphere, then it has a finite number of zeros and poles, and the sum of the orders of its poles equals the sum of the orders of its zeros. Every rational function is meromorphic on the whole Riemann sphere, and, in this case, the sum of orders of the zeros or of the poles is the maximum of the degrees of the numerator and the denominator.
Examples
The function
The function
The function
The function
All above examples except for the third are rational functions. For a general discussion of zeros and poles of such functions, see.
Function on a curve
The concept of zeros and poles extends naturally to functions on a complex curve, that is complex analytic manifold of dimension one. The simplest examples of such curves are the complex plane and the Riemann surface. This extension is done by transferring structures and properties through charts, which are analytic isomorphisms. More precisely, let be a function from a complex curve to the complex numbers. This function is holomorphic in a neighbourhood of a point of if there is a chart such that is holomorphic in a neighbourhood of Then, is a pole or a zero of order if the same is true for If the curve is compact, and the function is meromorphic on the whole curve, then the number of zeros and poles is finite, and the sum of the orders of the poles equals the sum of the orders of the zeros. This is one of the basic facts that are involved in Riemann–Roch theorem.
