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Numerical Evaluation of the Integral Occurring in the Theodorsen Arbitrary Airfoil Potential Theory

A more precise method of evaluating the e-integral
occurring in the arbitrarv airfoil theory of Theodorsen
(UACA Reps. Fos. 411 and 452) is developed by retaining
higher order terms in the Taylor expansion and by use
of Simpson's rule. Formulas are given for routine GEIGUP
lation of the 6—integral and for the necessary compu—
tational coefficients. The computational coefficients
are tabulated for a 40-point division of the range of
integration from 0 to 2H. With no increase in compu—
tational work the systematic error in the numerical
value of e is reduced from the order of 1 percent to
approximately 0.1 percent.

The solution of the general problem by means of
conformal transformation for the flow about an arbitrary
airfoil (references 1 and 2), a symmetrical lattice
(reference 5), and a biplane (reference 4) involves the
determination of the imaginary part of a complex trans-
formation function, given the real part. As shown in
references 1 and 2 the real part may be expanded in a
Fourier series and the imaginary part is the conjugate
Fourier series.

This integral occurs frequently in conformal—
transformation problems involving the evaluation of the
functions on the circle. A procedure for the numerical
evaluation of this integral is given in references 1
and 2. This method, which is currently in use at LMAL,
gives an error of about 1.5 percent for a 40-point
division of the range of integration. An improvement
in the accuracy is therefore very desirable, particularly
if the labor involved is not increased. A revised method
given herein is found to involve a little less work than
that previously required and to give an error of only
about 0.1 percent. Constants for use in this more pre-
cise method have been computed and are presented in
table I.