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This is a preview of the volume eight of  'Critical Comments ...'         
    


                         C R I T I C A L   C O M M E N T S   O N

                                WHY BOTHER TO COMPUTE?
                                           and related papers

                                              
                                               P A R T   V I I I

                               Complex Variables in Engineering






                                        Compiled by: T. Leko

------------------------------------

Part  VIII  is a continuation of the Critical  Comments*
series compiled in 1988  (Part I), 1989  (Part II),
1990  (Part III),  1992  (Part IV),  1993  (Part V),  
1994  (Part VI), and 1995  (Part VII)  on
 
                    WHY BOTHER TO COMPUTE?
                               and related papers

The present volume further advances the problems in
the theory of analytic functions and in particular in the
use of complex variables in engineering topics such as: 
                    Power series
                    Differential Equations
                    Contour Integrals
                    Boundary value problems
                    Problems of Limits
                    Questions of Analyticity
                    Complex Zeros 
and others.

3350 Martha Custis Drive                                                  
 T. Leko
Alexandria, VA 22302-2115                                   December 1996
E-Mail: tleko@aol.com
Phone-FAX:  +1-703-998-8093

*LCCN 91-184909
 QA431.C75 1988

------------------------------------ 





      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
 
     YOUR COMMENTS WILL BE GREATLY APPRECIATED !
 
 
---------------------------------------
 




                           COMMENTS BY:

                           D.  GRAYSON

                           T.   HOLLEBEEK

                           M.  MCCARTY

                           D.  ULLRICH

                             .
                             .
                             .
                             .

----------------------------------




                           D. GRAYSON

----------------------------------                      

Subj: sci.math.research article titled "several complex variables"
Date:  96-07-01  14:19:03 EDT
From: dan@math.uiuc.edu (Dan Grayson)
Reply-to:sci-math-research-request@uiuc.edu
To:  	tleko@aol.com


Sci.math.research is devoted to discussions of current mathematical
research, and as moderator of it, I must accept or reject each article
for release to the net.  I am returning your posting to you because
its subject material falls outside our charter.  You might consider
posting it to sci.math instead.  If this article is crossposted to several
newsgroups (look at the Newsgroups line below) then you will have to
repost
the article to the other ones.

The readers of sci.math.research may even be the best audience for your
query, but I can't consider that when deciding whether your article's
subject matter falls within our charter.

Please don't regard this rejection as any indication that your posting 
is in any way frivolous, annoying, or uninteresting to our readers, or
that 
you have in any way annoyed or wasted the time of the moderator.  Our news
group has an intentionally narrow charter, and there is no automatic
mechanism for those who post articles to learn what it is.

We welcome submissions from new and regular readers alike.  Although we
cannot accept this posting, we encourage you to try again in the future.

-------------------------------


                                               T.  HOLLEBEEK

---------------------------------

Subj:   Re: Bracket a complex root
Date:   96-07-06 12:48:44 EDT
From:  tim@franck.Princeton.EDU (Tim Hollebeek)
To:      Tleko@aol.com

Tleko@aol.com writes ...
> 
>      Example 1.  The analytic property of:
>  
>                  f(z) = 1/(1-z)^2
>                                            
>                  (R. Mennicken)

[...]

>      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.

Only people with a very poor knowledge of complex analysis have such
an expectation, as example one is obviously analytic everywhere
_except_ z = 1 from basic principles of complex analysis.  People who
actually know complex analysis know that it also explains how to
handle the pole of order 2 at z = 1.  I hope to see a retraction from
you after you reread an introductory complex analysis text, since you
are obviously either greatly misguided or trying to fraudulently
obtain research funds.

--------------------------------------------------------------------------
-
Tim Hollebeek         | Disclaimer :=> Everything above is a true
statement,
Electron Psychologist |                for sufficiently false values of
true.
Princeton University  | email: tim@wfn-shop.princeton.edu
----------------------| http://wfn-shop.princeton.edu/~tim (NEW!
IMPROVED!)


Subj: Re: Bracket a complex root
Date: 07/06/96
From: tleko@aol.com
To: tim@wfn-shop.priceton.edu
CC: dan@math.uiuc.edu

Dear Professor Hollebeek,

Thank you very much for you comments of July  6 , 1996.

I regret, however, that you have not read the letters by 
Professor Mennicken and others. 
If you have a FAX number or a mailing address I will gladly
send you the relevant references.

Your comments will be greatly appreciated.

With best regards, 
Toma Leko

--------------------------------------------


Subj: Re: Bracket a complex root
Date: 96-07-06 17:03:54 EDT
From: tim@franck.Princeton.EDU (Tim Hollebeek)
To: Tleko@aol.com

 However this shouldn't be necessary, as my comments regarded things
 that are taught to sophomore Math majors, which you certainly are
 aware of.  I would appreciate specific answers to my comments 
(as opposed to an obvious form letter) at your earliest convenience 
(assuming, of course, you are not involved in some fraudulent attempt 
to procure research funding like you appear to be).

Thank you for your time.

--------------------------------------------------------------------------
-
Tim Hollebeek         | Disclaimer :=> Everything above is a true
statement,
Electron Psychologist |                      for sufficiently false values
of true.
Princeton University  | email: tim@wfn-shop.princeton.edu
----------------------| http://wfn-shop.princeton.edu/~tim (NEW!
IMPROVED!)   

--------------------------------------

Subj: Re: Bracket a complex root
Date: 07/07/96
To: tim@wfn-shop.princeton.edu, tim@franck.princeton.edu
CC: dan@math.uiuc.edu,
 nleko@ceme.umd.edu

 Dear Professor Hollebeek,

 To respond specifically to your challenge to the results by Professors
 Mennicken, Goerisch and Habetha ( documented in the seven volumes 
 of 'Critical Comments ..' ) I offered to deliver to you all the relevant 
 documents. Please indicate your FAX number or a mailing address.

Best regards, Toma

--------------------------------------


Subj: Re: Bracket a complex root
Date: 96-07-07 11:36:35 EDT
From: tim@franck.Princeton.EDU (Tim Hollebeek)
To: Tleko@aol.com

Surely it takes less than 7 volumes to respond to a one line
criticism. The fact that you continually dodge such a simple
question leads me to believe you want to send me 7 volumes of
bullshit which would be a waste of my time, hence I am not
interested.

I hope you won't bother me further unless you plan on being
more forthcoming and honest.

--------------------------------------------------------------------------
-
Tim Hollebeek | Disclaimer :=> Everything above is a true statement,
Electron Psychologist | for sufficiently false values of true.
Princeton University | email: tim@wfn-shop.princeton.edu
----------------------| http://wfn-shop.princeton.edu/~tim (NEW!
IMPROVED!)
--------------------------------------

Subj: Re: Bracket a complex root
Date: 07/07/96
To: tim@wfn-shop.princeton.edu, tim@franck.princeton.edu
CC: dan@math.uiuc.edu, anthony@ynot1.demon.co.uk,
CC: hbaker@netcom.com, kovarik@mcmail.cis.mcmaster.ca,
        hawhorn@waikato.ac.nz,
        t762@@inforamp.net, nleko@ceme.umd.edu

Dear Professor Hollebeek,

  Not being interested in the subject I certainly will
 not bother you anymore.

 With best regards, Toma Leko

Note: To view the zeros of a complex function take 
the real (10) and the imaginary part (11) to run MATLAB's
surface commands. Several such plots are included in the
shown reference.

--------------------------------
Subj:	Re: Critical Comments ...
Date:	96-07-14 12:00:26 EDT
From:	tim@franck.Princeton.EDU (Tim Hollebeek)
To:	Tleko@aol.com

Tleko@aol.com writes ...
> 
> This is a preview of the volume eight of  'Critical Comments ...'

Hmm, a couple of weeks ago after being unable to answer simple
undergraduate math questions, you promised to no longer bother me with
your delusions.  Is your memory inadequate as well?

Also, I retain the copyright to my mailings to you, and have not
granted you permission to use them.  Please remove any reference to me
from your work.

-Tim Hollebeek

------------------------------------

Subj: Re:Copyrights
Date: 07/14/96
To: tim@wfn-shop.princeton.edu
CC: chappell@spica.math.uiuc.edu

Dear Tim,

What is the problem with you ?

What do you know about copyrights ?

Please let me know.

Best regards, Toma

------------------------------------------





                                  M.  MCCARTY

---------------------------------------

Subj: Re: Bracket a complex root
Date: 96-07-08 12:15:18 EDT
From: JMCCARTY@altair.dsccc.com (MIKE MCCARTY -- 93789)
To: Tleko@aol.com
CC: JMCCARTY@altair.dsccc.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.
) 

[elementary stuff cut]

)                      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
) 
)     YOUR COMMENTS WILL BE GREATLY APPRECIATED !

What are these supposed to be examples of?

f(z) = 1/(1-z)^2 is -not-  analytic on the unit disc. I wasn't aware
anyone expected it to be. =I= am certainly not shocked at it. It
obviously has a pole of second order at z = 1.

The function

      f(z) = ((1-z)/(pi sqrt z)) ln((1+sqrt z)/(1-sqrt z))

is singular at z = 0. The derivative (unless I made a mistake in
calculation) is 
	               
      f'(z)= -( (2t^3-t^2+1)/(pi t^2))ln((1+t)/(1-t))+1/(pi t)

where t = z^(1/2).

As one can easily see, the only place where a singularity is possible
is where t = 0 (and hence z = 0) or t = -1 (and z = +/- i), or t = +1
(and z = +/- 1). In the first case, the limit of the derivative is 0,
but the derivative does not actually exist there. Whether this is a
removable singularity, I don't know (the work looks like low return for
effort expended to me). For t = -1, one gets a logarithmic singularity
in the derivative. Hence the function fails to be analytic on the unit
disc. For t = 1, one again gets a logarithmic singularity. So for at
least four points on the unit disc, the function is not analytic, as
expected. This is by standard analysis.

What is the path of integration supposed to be for the third example?

Frankly, I don't understand what you are getting at.

Mike
----
char *p="char
*p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}

I do not speak for DSC. <- They make me say that.

----------------------- 

Subj: Bracket a complex root
Date: 07/08/96
From: tleko@aol.com
To:   jmccarty@altair.dsccc.com

Dear Professor McCarty,

Thank you very much for your letter of July 8, 1996. 

May you please give me your mailing address to send you 
the volume seven of "Critical Comments ..' which I
presented in Prague. You will have it in a day or two
by priority mail.

Your comments will be greatly appreciated.

Best regards, Toma Leko

----------------------------------

Subj: RE: Bracket a complex root
Date: 96-07-08 17:57:39 EDT
From: JMCCARTY@altair.dsccc.com (MIKE MCCARTY -- 93789)
To: Tleko@aol.com
CC: JMCCARTY@altair.dsccc.com 

You are welcome for the letter. I am not a professor. I am a senior
level software engineer. My undergraduate degree is in mathematics, my
graduate work was in mathematical probability and statistics (mostly
measure theory on sigma algebras of sets, really). I taught at the
University of Texas at Dallas while I was a graduate student.

But I would be glad to look over whatever.

Mike McCarty
2203 Royal Oaks Drive
Irving, Texas TPC 75060
----
char *p="char
*p=%c%s%c;main(){printf(p,34,p,34);}";main(){printf(p,34,p,34);}

I do not speak for DSC. <- They make me say that.
Mike
-----------------------------------------


                             D.  ULLRICH

-----------------------------------------

Subj: Re: Critical Comments ...
Date: 96-07-14 10:49:42 EDT
From: ullrich@hardy.math.okstate.edu
Sender: ullrich@hardy.math.okstate.edu
To: Tleko@aol.com

Why are you sending me this idiocy? Don't.
DU
------------------------------- 
Dear Dr. Ullrich,               July 14, 1996

Thank you very much for your contribution.

Best regards,

T. Leko
-------------------------------