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From: Tleko@aol.com 

      Presented at the 1996 GAMM-Conference in Prague
    
      Leko, T.
 
      COMPLEX VARIABLES IN ENGINEERING TRUE OR FALSE?
 
      Abstract: This paper disproves the foremost
      indispensible fundamental importance of Cauchy's
      integrals in the theory of analytic functions of
      a complex variable. In particular, Cauchy's
      integral theorem, Cauchy's integral formula,
      the definition of analyticity, Taylor's series
      and Laurent's series have all escaped the
      desired scrutiny.
 
                         1. Complex Analysis
     
     Complex analysis is recognizable by imaginary number
 
     (1)                    i
 
     which is either
    
     (2)                    i = sqrt(-1)
 
     or
 
     (3)                    i = (ln(-1))/pi   .

     
     Representation of complex numbers
 
     (4)                    z = x+iy
         
     is either in the orthogonal coordinate system or alternativaly in 
     geometric representation
 
     (5)                    z = R exp(i theta)
     
     where
 
     (6)                    R =  sqrt (x^2+y^2)
     
     (7)                    theta = arctan(y/x)  .
     
 
                            2. Complex Functions
 
     With z = x+iy or z = R exp(i theta) there are complex functions such
     as
 
     (8)                    f(z) = P(x,y)+iQ(x,y)
     
     where P(x,y) is the real and Q(x,y) the imaginary part. In geometric
     representation we have
 
     (9)        f(z)=sqrt(P(x,y)^2+Q(x,y)^2) exp(iarctan(Q(x,y)/P(x,y))
     
     where
 
     (10)   Real f(z) =sqrt(P(x,y)^2+Q(x,y)^2) cos(arctan(Q(x,y)/P(x,y)))
     
     (11)   Imag f(z) = sqrt(P(x,y)^2+Q(x,y)^2)sin(arctan(Q(x,y)/P(x,y)))


     
                3. Applications of Complex Analysis in Engineering
 
                              Power Series
                              Differential Equations
                              Contour  Integrals
                              Boundary Value Problems
                              Problems of Limits
                              Questions of Analyticity
 
 
                      4. Three Examples

     Example 1.  The analytic property of:
 
                 f(z) = 1/(1-z)^2
                                           
                 (R. Mennicken)
 
     Example 2.  On a boudary value problem via Poisson's integral:
 
                 f(z) = ((1-z)/(pi sqrt z)) ln((1+sqrt z)/(1-sqrt z))
 
                 (F. Goerisch)

     Example 3.  On the value of Cauchy's integral:
 
                 Int (1/(t-z)) dt
 
                 (K. Habetha)  .

 
                       5. Results
 
     The results of the three examples are documented in the seven
     volumes of 'Critical Comments on Why Bother to Compute? catalogued
     in the Library of Congress, Washington, DC and several university
     libraries in the USA and Germany. In their private communications
     Professors Mennicken, Goerisch and Habetha have verified that the
     three examples do not represent analytic functions in the unit
     disk contrary to expectation.
 
     Address: T. Leko, Independent Research Press, 3350 Martha Custis
              Drive, Alexandria, VA 22302-2115 USA, Phone-Fax:
              + 1 703 998 8093, E-mail: tleko@aol.com
 
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