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”.
How do you prove a function is meromorphic?
- A function on a domain Ω is called meromorphic, if there exists a sequence of points p1,p2,··· with no limit point in Ω such that if we denote Ω∗ = Ω \ {p1,···} • f : Ω∗ → C is holomorphic. •
- To see this, note that F(1/z) has either a pole or zero at z = 0. In either.
- Pk. ( 1.
- Pk ( 1 z − pk ) = 0.
What is transcendental meromorphic function?
There is a transcendental meromorphic function such that F(f) has a sequence of multiply-connected components Ai , i ∈ N, all different, such that each Ai separates 0 and ∞ and f(Ai) ⊂ Ai+1 , i ∈ N. Moreover A2i → ∞ as i → ∞ and A2i+1 → 0 as i → ∞.
What is difference between holomorphic and meromorphic?
Definitions: Holomorphic and Meromorphic A function that is analytic on a region A is called holomorphic on A. A function that is analytic on A except for a set of poles of finite order is called meromorphic on A.
Are meromorphic functions continuous?
A meromorphic function in is defined as a global section of , i.e. a continuous mapping such that for all .
Is an entire function meromorphic?
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.
What is meant by removable singularity?
In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourhood of that point.
What is the difference between a meromorphic and entire function?
Are all holomorphic functions meromorphic?
Every holomorphic function is meromorphic, but not vice versa.
Which of the following is entire function?
Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error …
What are Laurent series used for?
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
Is removable singularity a pole?
singularities. …it is known as a removable singularity. In contrast, the above function tends to infinity as z approaches 0; thus, it is not bounded and the singularity is not removable (in this case, it is known as a simple pole).
How do you prove that an entire function is meromorphic in C?
Let fbe meromorphic in C. Then there are entire functions gand hsuch that f= g=h. Proof. Let (z j) be the (\fnite or in\fnite) sequence of poles of fin Cnf0g, counted according to multiplicity. By Theorem 8.3 there exists an entire function kwhich has precisely the z jas zeros. If 0 is a pole of f, let mbe its multiplicity and put m= 0 otherwise.
Are there any meromorphic functions with four totally different values?
There are also meromorphic functions with four totally rami\fed values. They come from the theory of elliptic (doubly periodic) functions. 13 Uniqueness theorems Our \frst application of the second fundamental theorem are some uniqueness the- orems. De\fnition 13.1. Let fand gbe meromorphic and a2Cb.
What is Mero- morphic function?
eiz+ eiz : A function which is holomorphic in a domain Dexcept for poles is called mero- morphic in D. (The precise de\fnition and more details will be given later.) In particular if D = C we just say the function is meromorphic.
How do you prove that a transformation is meromorphic?
Let f be meromorphic and M a Mobius transformation. Then ˆ(Mf) = ˆ(f) and (Mf) = (f). The proof follows directly from Theorem 5.2. Theorem 7.4. Let f 1and f 2be meromorphic.