Genz Corner Peak integral

745 days ago by Volpe

var('C_x, C_y, R, x, y, a, b, c, d') f = 1/(1+C_x * x +C_y * y)**(2 + R) f 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\left(C_{x} x + C_{y} y + 1\right)}^{R + 2}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\left(C_{x} x + C_{y} y + 1\right)}^{R + 2}}
forget() (C_x*x+b*C_y+1 > 0).assume() (C_x*x+1 > 0).assume() (R > 0).assume() (b > 0).assume() (d > 0).assume() (C_x > 0).assume() (C_y > 0).assume() assumptions() 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[C_{x} x + C_{y} b + 1 > 0, C_{x} x + 1 > 0, R > 0, b > 0, d > 0, C_{x} > 0, C_{y} > 0\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[C_{x} x + C_{y} b + 1 > 0, C_{x} x + 1 > 0, R > 0, b > 0, d > 0, C_{x} > 0, C_{y} > 0\right]
F = f.integral(y, 0, b).integral(x, 0, d) F.expand() 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{-e^{\left(-R \log\left(C_{y} b + 1\right)\right)}}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}} + \frac{-e^{\left(-R \log\left(C_{x} d + 1\right)\right)}}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}} + \frac{e^{\left(-R \log\left(C_{x} d + C_{y} b + 1\right)\right)}}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}} + \frac{1}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{-e^{\left(-R \log\left(C_{y} b + 1\right)\right)}}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}} + \frac{-e^{\left(-R \log\left(C_{x} d + 1\right)\right)}}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}} + \frac{e^{\left(-R \log\left(C_{x} d + C_{y} b + 1\right)\right)}}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}} + \frac{1}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}}
F_x = f.integral(y, 0, b).integral(x, 0, infinity) F_x.expand().simplify() 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{-e^{\left(-R \log\left(C_{y} b + 1\right)\right)}}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}} + \frac{1}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{-e^{\left(-R \log\left(C_{y} b + 1\right)\right)}}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}} + \frac{1}{{\left(C_{x} C_{y} R^{2} + C_{x} C_{y} R\right)}}
F_y = f.integral(y, 0, infinity).integral(x, 0, d) F_y.expand().simplify() 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\left(R + 1\right)} C_{x} C_{y} R} + \frac{-1}{{\left(R + 1\right)} {\left(C_{x} d + 1\right)}^{R} C_{x} C_{y} R}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\left(R + 1\right)} C_{x} C_{y} R} + \frac{-1}{{\left(R + 1\right)} {\left(C_{x} d + 1\right)}^{R} C_{x} C_{y} R}
F_xy = f.integral(y, 0, infinity).integral(x, 0, infinity) F_xy 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\left(R + 1\right)} C_{x} C_{y} R}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{1}{{\left(R + 1\right)} C_{x} C_{y} R}