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Conference rusure::math

Title:Mathematics at DEC
Moderator:RUSURE::EDP
Created:Mon Feb 03 1986
Last Modified:Fri Jun 06 1997
Last Successful Update:Fri Jun 06 1997
Number of topics:2083
Total number of notes:14613

1100.0. "Complex Exponents" by DRUMS::FEHSKENS () Fri Jul 14 1989 18:52

    Last month's Scientific American includes a discussion of "biomorphs"
    generated by a Mandelbrot-set-like iteration process.  Some of the
    more interesting forms involve raising complex numbers to complex
    powers, specifically complex "self exponentiation", i.e., z^z with
    z = x + i*y.
    
    I can do the obvious (x + i*y)^(x + i*y) = ((x + i*y)^x)*((x + i*y)^i*y) 
    and can then do (x + i*y)^x, but I don't know (and can't find in
    any of my immediately available math books) how to raise a complex
    number to an imaginary power.
    
    Is there something special about the "self exponentiation" case?
    Can somebody explain how to do complex exponentiation in general?
    Any help or pointers appreciated.
    
    len.
T.RTitleUserPersonal
Name		DateLines
1100.1Some helpful identitiesPOOL::HALLYBThe Smart Money was on GoliathFri Jul 14 1989 20:066
1100.2RDVAX::NGSat Jul 15 1989 00:587
    You will also need this:
    
    Ln z = ln |z| + i*arg(z);       z = x+iy
    
    where the second 'ln' is the natural logarithm for the real number,
    |z| = (x^2+y^2)^0.5 and 'arg' stands for the angle of the complex
    number z.
1100.3AITG::DERAMODaniel V. {AITG,ZFC}:: D'EramoSat Jul 15 1989 02:3313
        Write a + bi as re^(it)         [that's "t" for "theta"]
        where r^2 = x^2 + y^2 and e^(it) = cos t + i sin t, and
        the "principal value" (or is it "principle value"?) of t
        is used.
        
        Then (a + bi)^(c + di) = (re^(it))^(c + di)
        	= (e^(ln r + it)) ^ (c + di)
                = e^((ln r + it)(c + di))
        	= e^( c ln r - dt + (d ln r + ct)i )
        	= (e^x)(cos y + i sin y)  where x = c ln r - dt
        					y = d ln r + ct
        
        Dan
1100.4with other functions tooANT::JANZENcf. ANT::CIRCUITS,ANT::UWAVESMon Jul 17 1989 16:353
    This is all defined in my complex arithmetic package for Ada in the
    Ada toolshed conference.
Tom
1100.5ThanksDRUMS::FEHSKENSWed Jul 19 1989 15:287
    This all looks familiar, I just wasn't sure I could do exponentials
    the same way with complex numbers as reals.
    
    Thanks for the confirmation.
    
    len.