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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>