Finite Difference Cauchy-Riemann Operators and Their Fundamental Solutions in the Complex Case

2003 ◽  
pp. 101-115 ◽  
Author(s):  
Klaus Gürlebeck ◽  
Angela Hommel
Keyword(s):  
Finite Difference ◽  
Fundamental Solutions ◽  
Complex Case ◽  
Cauchy Riemann

Universal Series on a Riemann Surface

2011 ◽  
Vol 63 (5) ◽  
pp. 1025-1037
Author(s):  
Raphäel Clouâtre
Keyword(s):  
Riemann Surface ◽  
Holomorphic Function ◽  
Compact Subset ◽  
Fundamental Solutions ◽  
Partial Sums ◽  
Universal Series ◽  
Cauchy Riemann ◽  
Series Of Functions

Abstract Every holomorphic function on a compact subset of a Riemann surface can be uniformly approximated by partial sums of a given series of functions. Those functions behave locally like the classical fundamental solutions of the Cauchy–Riemann operator in the plane.

scholarly journals On the complex q-Appell polynomials

2020 ◽  
Vol 74 (1) ◽  
pp. 31
Author(s):  
Thomas Ernst
Keyword(s):  
Imaginary Part ◽  
Generating Function ◽  
Complex Number ◽  
Analytic Functions ◽  
Complex Case ◽  
Euler Polynomials ◽  
Appell Polynomials ◽  
The Real ◽  
Cauchy Riemann

The purpose of this article is to generalize the ring of \(q\)-Appell polynomials to the complex case. The formulas for \(q\)-Appell polynomials thus appear again, with similar names, in a purely symmetric way. Since these complex \(q\)-Appell polynomials are also \(q\)-complex analytic functions, we are able to give a first example of the \(q\)-Cauchy-Riemann equations. Similarly, in the spirit of Kim and Ryoo, we can define \(q\)-complex Bernoulli and Euler polynomials. Previously, in order to obtain the \(q\)-Appell polynomial, we would make a \(q\)-addition of the corresponding \(q\)-Appell number with \(x\). This is now replaced by a \(q\)-addition of the corresponding \(q\)-Appell number with two infinite function sequences \(C_{\nu,q}(x,y)\) and \(S_{\nu,q}(x,y)\) for the real and imaginary part of a new so-called \(q\)-complex number appearing in the generating function. Finally, we can prove \(q\)-analogues of the Cauchy-Riemann equations.

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