Re: Integration problem
- To: mathgroup at smc.vnet.net
- Subject: [mg40620] Re: [mg40577] Integration problem
- From: Vladimir Bondarenko <vvb at mail.strace.net>
- Date: Fri, 11 Apr 2003 02:03:11 -0400 (EDT)
- References: <200304100741.DAA24290@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
Thursday, April 10, 2003, 4:41:35 AM, Stefano Fricano <stefanofricano at yahoo.it> wrote: SF> i.e. Integrate[( A /(A^2-B^2)^0.5 )*((A+d) / ((A+d)^2-B^2)^0.5 )) SF> between B and B + C SF> Any idea? Let's rewrite Integrate[(A/(A^2 - B^2)^(1/2))*((A + d)/((A + d)^2 - B^2)^(1/2)), {A, B, B + C}] as i1 + i2 where A = z, B = b, C = c, i1 = Integrate[z^2/(Sqrt[z^2 - b^2]*Sqrt[z^2 + 2*d*z - b^2 + d^2]), {z, b, b + c}]; i2 = d*Integrate[z/(Sqrt[z^2 - b^2] Sqrt[z^2 + 2 d z - b^2 + d^2]), {z, b, b + c}]; answer = FullSimplify[i1+i2] (c*(b*Sqrt[(c*(-2*b + d))/((2*b + c)*d)]*(c + d)*(2*b + c + d) + (2*b + c)*Sqrt[(b*(c + d))/((2*b+ c)*d)]*Sqrt[(b*(2*b + c + d))/ ((2*b + c)*(2*b + d))]*((4*b^2 - d^2)*EllipticE[ArcSin[Sqrt[(c*( -2*b + d))/((2*b + c)*d)]],d^2/(-4*b^2 + d^2)]- 2*b^2*EllipticF[ ArcSin[Sqrt[(c*(-2*b + d))/((2*b+c)*d)]],d^2/(-4*b^2 + d^2)])))/ (b*Sqrt[c*(2*b + c)]*Sqrt[(c*(-2*b+ d))/((2*b + c)*d)]*Sqrt[(c + d)*(2*b + c + d)]) Let's check the answer at random. answer /. {b -> 1, c -> 1, d -> 1} // N // Chop NIntegrate[(z/(z^2-1)^(1/2))*((z+1)/((z+1)^2-1)^(1/2)),{z,1,2}] 1.92102 1.92102 answer /. {b -> 2, c -> 2, d -> 2} // N // Chop NIntegrate[(z/(z^2-2^2)^(1/2))*((z+2)/((z+2)^2-2^2)^(1/2)),{z,2,4}] 3.84204 3.84204 answer /. {b -> 1/2, c -> 1/3, d -> 1/4} // N // Chop NIntegrate[(z*(1/4+z))/(Sqrt[-1/4+z^2]*Sqrt[-1/4+(1/4+z)^2]),{z,1/2,5/6}] 0.825424 0.825424 That's not enough of cause; please analyze its behavior, e.g. find the asymptotics over b, c, d, substitute more choices etc A remark for Liana: you must be perfect with such stuff to enter my project at all. Best wishes, Vladimir Bondarenko Mathematical and Production Director Symbolic Testing Group Web : http://www.CAS-testing.org/ GEMM Project (95% ready) Email: vvb at mail.strace.net Voice: (380)-652-447325 Mon-Fri 6 a.m. - 3 p.m. GMT Mail : 76 Zalesskaya Str, Simferopol, Crimea, Ukraine
- References:
- Integration problem
- From: stefano fricano <stefanofricano@yahoo.it>
- Integration problem