The classical Hadamard product of two functions f and g, holomorphic in open neighborhoods of zero, can be extended to a more general case, where the functions can be singular at zero. Then the extended product \(f*g\) can be naturally defined by multiplying coefficients of their Laurent expansions at zero. Next for given functions \(\lambda \) and \(\eta \) holomorphic in the complex plane with removed zero we consider the product \((\lambda f)*(\eta g)\) , called the Hadamard \(\lambda ,\eta \) -product of f and g. We show that if simply connected domains A and B contain zero, then the Hadamard \(\lambda ,\eta \) -product of functions f and g, holomorphic in \(A\setminus \{0\}\) and \(B\setminus \{0\}\) respectively, has a holomorphic extension to the connected component containing zero of the star product \(A*B\) provided zero is a regular point of the function \((\lambda f)*(\eta g)\) , and otherwise to the component with removed zero. The extension has an explicit integral representation which generalizes the familiar Parseval integral form of the classical Hadamard product. The obtained results are relevant to the Hadamard Multiplication Theorem.
