The generalization of the classical Hadamard theorem of multiplication of singularities onto multidimensional complex situation is connected with the construction of the Hadamard composition in \(\mathbb{C}^n\). Thus, the realization of analog of the Hadamard composition by use of the Szegö kernel of an arbitrary \(n\)-circular domain in \(\mathbb{C}^n\) is well known. But in the present paper the Hadamard composition of series \(f(z)= \sum_{\alpha\in Z^n_+} a_\alpha z^\alpha\) and \(g(z)= \sum_{\alpha\in Z^n_+} b_\alpha z^\alpha\) is understood to be the series \(h(z)= \sum_{\alpha\in Z^n_+} a_\alpha b_\alpha c_\alpha^{-1} z^\alpha\), where \(c_\alpha\) are the Taylor coefficients of the function \[ U{pq} (z)= \sum_{\alpha\in Z^n_+} \bigl(p |\alpha|+ pn-n+q-1\bigr)! \biggl(\bigl\{p\alpha+ (p-1)I \bigr\}! (n-1)!\biggr)^{-1} p^{1-n} z^\alpha \] \((I\) is the unit multi-index, \(p\in N\), \(q= \overline {1,n})\). In so doing, note that \(U_{pn}\) is the Szegö kernel of the domain \(D_p= \{z\in\mathbb{C}^n: \sum|z_k|^{2/p} < 1\}\). That is \(n\) different compositions (including generated by the Szegö kernel) can be constructed for the domains \(D_p\). In the first part of the article the notion of \(p\)-convexity introduced by the author allows to obtain a number of new results on the integral representations and analytic continuation of a composition as well as consider the question of expansion of the composition in the absolutely and uniformly convergent series of the special form. The second part of the paper is devoted to the ranges of the Hadamard compositions constructed both by the way suggested by the author and by the method, which was known formerly, -- with the help of the Szegö kernel of a complete \(n\)-circular domain.