1550251515-Classical_Complex_Analysis__Gonzalez_

Integration 517

We may consider also the fl.ow determined by a complex potential of

the form

F(z) = _J_ logz

·27f

where q = m + ik is a complex number with k -:f. 0. In this case the velocity

potential is given by

1

U = - ( m ln r + kB)

27f

and the stream function by

1

V =

2

7f ( mB - k ln r)

where z = reie. Hence the equipotential lines are the spirals

r = c1e-k(}/m

and the streamlines are the orthogonal spirals

r = Czem(}/k

In this case the fl.ow determined by the complex potential can be interpreted
as the superposition of a fl.ow with vertex at z = 0 of strength k, and a
fl.ow with source (or sink) of strength lml, also at z = 0. Clearly, the point
oo may also be considered as a spiral vortex.
A similar analysis can be used for the discussion of a number of two-
dimensional questions in gravitational theory, electrostatics, and steady-
state conduction of heat. In fact, the analysis is the same in the various
classes of problems, with appropriate changes in terminology. For further
details the student is referred to standard treatises on these subjects.

Bibliography

1. L. V. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1979.


2. T. M. Apostol, Mathematical Analysis, Addison-Wesley, Reading, Mass., 1957.


3. R. B. Ash, Complex Variables, Academic Press, New York, 1971.


4. N. C. Ankeny, One more proof of the fundamental theorem of algebra, Amer.
   Math. Monthly, 54 (1947), 464.


5. R. P. Boas, Yet another proof of the fundamental theorem of algebra, Amer.
   Math. Monthly, 71 (1964), 180.


6. S. Bochner, Functions in One Complex Variable as Viewed from the Theory
   of Functions in Several Variables, University of Michigan Press, Ann Arbor,
   Mich., 1955.