Table of Contents
Which of the function is a 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. The term comes from the Ancient Greek meros (μέρος), meaning “part”.
Is meromorphic function analytic?
In one complex dimension (one complex variable), hence on a Riemann surface, a meromorphic function is a complex-analytic function which is defined away from a set of isolated points.
Are meromorphic functions entire?
A function is said to be entire if it is analytic on all of C. It is said to be meromorphic if it is analytic except for isolated singularities which are poles.
Why do we use Laurent series?
The method of Laurent series expansions is an important tool in complex analysis. Where a Taylor series can only be used to describe the analytic part of a function, Laurent series allows us to work around the singularities of a complex function.
What is the difference between a meromorphic and entire function justify your answer?
What is the difference between a meromorphic and entire function?
What is the principal part of Laurent series?
The principal part at of a function. is the portion of the Laurent series consisting of terms with negative degree. That is, is the principal part of at . If the Laurent series has an inner radius of convergence of 0 , then has an essential singularity at , if and only if the principal part is an infinite sum.
How do you write a Laurent series of a complex function?
Laurent’s Series Formula
- ∑ n = 0 ∞ a n ( z − z 0 ) n. converges to the analytic function, when |z-z0| < r2.
- ∑ n = 1 ∞ b n ( z − z 0 ) n. converges to the analytic function, when |z-z0| > r1.
What’s the difference between holomorphic and analytic?
A function f:C→C is said to be holomorphic in an open set A⊂C if it is differentiable at each point of the set A. The function f:C→C is said to be analytic if it has power series representation. We can prove that the two concepts are same for a single variable complex functions.
Which function is holomorphic?
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cn.
Are all holomorphic functions meromorphic?
Every holomorphic function is meromorphic, but not vice versa.
Which functions are holomorphic?
A function is holomorphic on an open set U, if it is complex differentiable at every point of U. A function f is holomorphic at a point z0 if it is holomorphic on some neighbourhood of z0.
What is an example of meromorphic function?
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.
Is this function meromorphic in the whole complex plane?
Thus, this function is not meromorphic in the whole complex plane. However, it is meromorphic (even holomorphic) on . is not meromorphic on the whole complex plane, as it cannot be defined on the whole complex plane while only excluding a set of isolated points. is an accumulation point of poles and is thus not an isolated singularity.
What is the meromorphic field of a complex number?
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,
How do you find the ratio of meromorphic functions?
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. The gamma function is meromorphic in the whole complex plane.