مرکز منطقه ای اطلاع رساني علوم و فناوري - On the evaluation of double square integral in the (s<inf>1</inf>, s<inf>2</inf>) complex biplane

Abstract :

In this correspondence, a method of evaluating the double square integral $\\int_{0}^{\\infty} \\int_{0}^{\\infty} [h(t_{1}, t_{2})]^{2} dt_{1} dt_{2}$ $= {1 \\over (2\\pi j)^{2}} \\int_{-j \\infty}^{j \\infty} \\int_{-j \\infty}^{j \\infty} H(s_{1}, s_{2}) H(-s_{1}, -s_{2}) ds_{1} ds_{2}$ where $h(t_{1}, t_{2}) = L^{-1} [H(s_{1}, s_{2})]$ is discussed. The method of evaluation is based on using the double bilinear transformation (DBT) $s_{k} = (1 - z_{k})/(1 + z_{k}), k = 1, 2$ to reduce the above complex integral to a double integral in the complex ( $z_{1}, z_{2}$ ) bidisk, ${\\infty \\atop {\\sum \\atop {m=0}}} {\\infty \\atop {\\sum \\atop {n=0}}} [a(m, n)]^{2} = {1 \\over (2{\\pi}j)^{2}} {\\oint \\atop {|z1| = 1}} {\\oint \\atop {|z2| = 1}} A(z_{1}, z_{2})$ $\\cdot A(Z_{1}^{-1}, Z_{2}^{-1}) frac{dz_{1}}{z_{1}}frac{dz_{1}}{z_{2}}$ . There exist several methods for evaluating the above double integral. In this correspondence, we mention one promising method. Also, a matrix method is proposed for applying the "DBT" to a quotient of two-dimensional (2-D) polynomials.