PRINCIPAL VALUE + 3D NINTEGRATE

• To: mathgroup at smc.vnet.net
• Subject: [mg81514] PRINCIPAL VALUE + 3D NINTEGRATE
• From: djokic at phy.bg.ac.yu
• Date: Wed, 26 Sep 2007 06:44:35 -0400 (EDT)

```A big HI to everybody !!!

A problem I am faced with is about numerically solving principal value
of a 3D integral whose integrand contains an infinity number of
singularities on a surface. So, in order not to be explaining that in
detail, here my problem is:

\!\(\(Q[y_] := \(52.  y\)\/ArcTanh[\(2.  y\)\/5. ];\)\
[IndentingNewLine]
\(g[y_] := FindRoot[Q[x] == y, {x, \ 5. \ , \ 2. }];\)\
[IndentingNewLine]
\(R[y_] := Re[\(\(g[y]\)[\([1]\)]\)[\([2]\)]];\)\[IndentingNewLine]
\(J[x_, y_, z_] :=
4.92 \((Cos[x] + Cos[y] + Cos[z])\) + 3.88 \((Cos[\(x + y\)\/
2=2E ] +
Cos[\(y + z\)\/2. ] + Cos[\(z + x\)\/2. ])\);\)\n
\(j[x_, y_, z_] := 3.88 \((Cos[\(x -
y\)\/2. ] + Cos[\(y - z\)\/2. ] + Cos[\(z - x\)\/2. ])\);\)\n
\(W[x_, y_, z_] :=
Sqrt[\((39.942 + J[x, y, z] - j[x, y, z])\) \((39.942 - J[x, y,
z] - j[
x, y, z])\)];\)\[IndentingNewLine]
\(=CE=A9[K_, T_, =CF=89_] := \(\(2.  \(K\^2\)
R[T]\^3\)\/=CF=80\^3\) NIntegrate[\((\(\((39.942 - j[x, y,
\
z])\)*\((39.942 - j[x, y, z])\)\)\/W[x, y,
z])\) \(1\/\(=CF=89\^2 - 4.  \(R[T]\^2\) W[
x, y, z]\^2 +
0.00000000000000001\)\) Coth[\(R[T] W[x, y, z]\)\/\(1.44
T\)], {x, \(-=CF=80\), =CF=80}, {y, \(-=CF=80\), =CF=80}, {z, \=
(-=CF=80\), =CF=80}];\)\
\[IndentingNewLine]
s = Table[=CE=A9[K = 2. , T, =CF=89 = 50. ], {T, 1. , 152. , 2. }]\)

When obtaining the function =CE=A9[T] and drawing it depending on parameter
T, the data result achieved in such a way does not look so smooth,
particularly in the case when =CF=89 <200. Well, how to get this function
as smoothed as possible?

All the best,

Dejan Djokic

P=2E S. 0.00000000000000001 has ben put in the integration to avoid
singularities on the surface.

```

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