Meromorphic function

In the mathematical field of complex analysis, a meromorphic function on an open subset${\displaystyle D}$ of the complex plane is a function that is holomorphic on all of ${\displaystyle D}$except for a set of isolated points, which are poles of the function.^[1] The term comes from the Greekmeros (μέρος), meaning 'part'.^[a]

Every meromorphic function on ${\displaystyle D}$ can be expressed as the ratio between two holomorphic functions (with the denominator not constant 0) defined on ${\displaystyle D}$: 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 ${\displaystyle z}$ and the numerator does not, then the value of the function will approach infinity; if both parts have a zero at ${\displaystyle z}$, then one must compare the multiplicity of these zeros.

From an algebraic point of view, if the function's domain 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.

The terms holomorphic and meromorphic were introduced in 1875 by Charles Briot and Jean-Claude Bouquet, two of Augustin-Louis Cauchy's students. Holomorphic derives from the Greek ὅλος (hólos) meaning "whole", and μορφή (morphḗ) meaning "form" or "appearance" or "type", while meromorphic derives from μέρος (méros) meaning "part". A holomorphic function resembles an entire function ("whole") in a domain of the complex plane while a meromorphic function, resembles a rational fraction ("part") of entire functions in a domain of the complex plane.^[2]

In the 1930s, in group theory, the term meromorphic function (or meromorph) was used in a different sense: it meant a function from a group into itself that preserved the group operation.^[3] This language is obsolete, and the term endomorphism is now used for such a function.

Since poles are isolated, there are at most countably many for a meromorphic function.^[4] The set of poles can be infinite, as exemplified by the function $${\displaystyle f(z)=\csc z={\frac {1}{\sin z}}.}$$

By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted, multiplied, and the quotient ${\displaystyle f/g}$ can be formed unless ${\displaystyle g(z)=0}$ 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.

• All rational functions,^[4] for example $${\displaystyle f(z)={\frac {z^{3}-2z+10}{z^{5}+3z-1}},}$$ are meromorphic on the whole complex plane. Furthermore, they are the only meromorphic functions on the extended complex plane.
• The functions $${\displaystyle f(z)={\frac {e^{z}}{z}}\quad {\text{and}}\quad f(z)={\frac {\sin {z}}{(z-1)^{2}}}}$$ as well as the gamma function and the Riemann zeta function are meromorphic on the whole complex plane.^[4]
• The function $${\displaystyle f(z)=e^{\frac {1}{z}}}$$ is defined in the whole complex plane except for the origin, 0. However, 0 is not a pole of this function, rather an essential singularity. Thus, this function is not meromorphic in the whole complex plane. However, it is meromorphic (even holomorphic) on ${\displaystyle \mathbb {C} \setminus \{0\}}$.
• The complex logarithm function $${\displaystyle f(z)=\ln(z)}$$ليست ميرومورفية على كامل المستوى المركب، إذ لا يمكن تعريفها على كامل المستوى المركب مع استبعاد مجموعة من النقاط المعزولة فقط. ^{[ 4 ]}
• الوظيفة$${\displaystyle f(z)=\csc {\frac {1}{z}}={\frac {1}{\sin \left({\frac {1}{z}}\right)}}}$$ليست ميرومورفية في المستوى بأكمله، لأن النقطة${\displaystyle z=0}$^{[ 4 ]} هي نقطة تراكم للأقطاب، وبالتالي فهي ليست نقطة تفرد معزولة .
• الوظيفة$${\displaystyle f(z)=\sin {\frac {1}{z}}}$$كما أنها ليست ميرومورفية، حيث أن لها نقطة تفرد أساسية عند 0.

على سطح ريمان ، تقبل كل نقطة جوارًا مفتوحًا يكون ثنائي الشكل مع مجموعة جزئية مفتوحة من المستوى العقدي. وبذلك، يمكن تعريف مفهوم الدالة الميرومورفية لكل سطح ريمان.

عندما تكون D هي كرة ريمان بأكملها ، فإن حقل الدوال الميرومورفية هو ببساطة حقل الدوال الكسرية في متغير واحد على الحقل المركب، إذ يمكن إثبات أن أي دالة ميرومورفية على الكرة هي دالة كسرية. (هذه حالة خاصة مما يُسمى بمبدأ GAGA ).

لكل سطح ريمان ، تكون الدالة الميرومورفية هي نفسها الدالة الهولومورفية التي تُسقط على كرة ريمان والتي ليست دالة ثابتة تساوي ∞. وتتوافق الأقطاب مع تلك الأعداد المركبة التي تُسقط على ∞.

على سطح ريمان غير متراص ، يمكن تمثيل كل دالة ميرومورفية كحاصل قسمة دالتين هولومورفيتين (معرفتين عالميًا). في المقابل، على سطح ريمان متراص، تكون كل دالة هولومورفية ثابتة، بينما توجد دائمًا دوال ميرومورفية غير ثابتة.

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