NEC-LIST: Quick method on Complex Exponential Integration

I am not clear on just what the integral is. If it is
Integrate[Exp[-I k u]/u,{u,u1,u2}] then it can be written in terms of
exponential integrals or sine and cosine integrals (Si and Ci).
Methods for evaluating these are given in Abramowitz and Stegun,
Handbook of Mathematical Functions for real and complex arguments.

If, as Alexandre Kampouris suggests, u=Sqrt[rho^2+(z-z')^2] and the
integral is Integrate[Exp[-I k u]/u,{z,z1,z2}] then it is a
generalized exponential integral. The subroutine INTX in NEC2
evaluates this type of integral using adaptive Romberg integration,
but it is not the most efficient method. NEC-4 uses a series
approximation, assuming k*Abs[z2-z1] is small. You can write the
integral as Exp[-I k u0]*Integrate[Exp[-I k(u-u0)]/u,{z,z1,z2}] where
u0 is the average of u at z=z1 and z=z2. Then use the small argument
approximation for Exp to four or five terms, collect terms in u^n and
integrate each term analytically. The form shown in the NEC-4 manual
results in cancellation of large numbers when k*u0 gets large. One
way around this problem is to use a three-point Gaussian integration
at large distances, as mentioned in the manual. However, it is faster
to develop a series for Abs[z2-z1] << u0. Then only one evaluation of
Exp and some algebra are needed.

Doug Werner at Penn State U. has published a lot of material that
includes evaluation of integrals of Exp[-I k R]/R^n as part of his
exact kernel evaluation. One paper is in IEEE Transactions APS,
Vol. 41, August 1993, but there are others including the ACES Journal
that may have more on the integrals.

The series used in NEC-4 is given in Appendix C of the NEC-4 Theory
Manual. I think this manual is available to the public at
http://www.llnl.gov under Publications, Online Public Access Catalog.
The NEC-4 User's Manual is not available to the public, but the Theory
manual apparently is. However, I cannot check this, since I am
accessing it from inside LLNL. The NEC-4 Theory Manual is call number
231654.

[Yes, the theory manual *is* accessible from outside LLNL - mod.]

Also, the complete NEC-2 manual is now available from www.llnl.gov
Publications, so this provides another option to the ones that have
been scanned by NEC users. The NEC-2 manual has call number 186099.

Jerry Burke
LLNL
Received on Thu Oct 15 1998 - 09:48:17 EDT