Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated points, which are poles of the function. This terminology comes from the Ancient Greek meros (μέρος), meaning 'part', as opposed to holos (ὅλος), meaning 'whole.' In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all of D except for a set of isolated points, which are poles of the function. This terminology comes from the Ancient Greek meros (μέρος), meaning 'part', as opposed to holos (ὅλος), meaning 'whole.' Every meromorphic function on D can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on D: any pole must coincide with a zero of the denominator. Intuitively, a meromorphic function is a ratio of two well-behaved (holomorphic) functions. Such a function will still be well-behaved, except possibly at the points where the denominator of the fraction is zero. If the denominator has a zero at z and the numerator does not, then the value of the function will approach infinity; if both parts have a zero at z, then one must compare the multiplicity of these zeros. From an algebraic point of view, if D is connected, then the set of meromorphic functions is the field of fractions of the integral domain of the set of holomorphic functions. This is analogous to the relationship between the rational numbers and the integers. In the 1930s, in group theory, a meromorphic function (or meromorph) was a function from a group G into itself that preserved the product on the group. The image of this function was called an automorphism of G. Similarly, a homomorphic function (or homomorph) was a function between groups that preserved the product, while a homomorphism was the image of a homomorph. This terminology is now obsolete. The term endomorphism is now used for the function itself, with no special name given to the image of the function. The term meromorph is no longer used in group theory. Since the poles of a meromorphic function are isolated, there are at most countably many. The set of poles can be infinite, as exemplified by the function By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient f / g {displaystyle f/g} can be formed unless g ( z ) = 0 {displaystyle g(z)=0} on a connected component of D. Thus, if D is connected, the meromorphic functions form a field, in fact a field extension of the complex numbers. In several complex variables, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, f ( z 1 , z 2 ) = z 1 / z 2 {displaystyle f(z_{1},z_{2})=z_{1}/z_{2}} is a meromorphic function on the two-dimensional complex affine space. Here it is no longer true that every meromorphic function can be regarded as holomorphic function with values in the Riemann sphere: There is a set of 'indeterminacy' of codimension two (in the given example this set consists of the origin ( 0 , 0 ) {displaystyle (0,0)} ). Unlike in dimension one, in higher dimensions there do exist complex manifolds on which there are no non-constant meromorphic functions, for example, most complex tori.