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MathGroup Archive 2004

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Re: proof of formula for log(-t) found in Mathematica?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg48217] Re: [mg48194] proof of formula for log(-t) found in Mathematica?
  • From: Raul Martinez <rmartinez at vrinc.com>
  • Date: Tue, 18 May 2004 04:16:41 -0400 (EDT)
  • References: <200405170722.DAA29602@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

Roger L. Bagula wrote:

>I found this while doing work on complex exponents:
>
>f(t)=Log[-t]=Log[t^2]/2+I*Arg[-t]
>
>It is a result built into Mathematica.
>I would like to see how it is derived as it seem counter intuitive in 
>it's results.
>Respectfully,
>Roger L. Bagula
>
>  
>
Roger,

If t is real, then

Log[-t] = Log[Exp[I Pi] |t|] = Log[|t|] + Log[Exp[I Pi]] = Log[|t|] + I Pi.

If t is complex, then we can write

-t = -a - I b = |t| Exp[I Arg[t]] = (a^2 + b^2)^(1/2) Exp[ I Arg [-a - I b]]

Thus,

Log[-t] = Log[(a^2 + b^2)^(1/2)] + Log[Exp[ I Arg [-a - I b]]]

= (1/2) Log[(a^2 + b^2)] + I Arg[-a - I b].

The above relation is consistent with

Log[t^2]/2+I*Arg[-t]

since

t^2 = t Conjugate[t] = (-a - I b)(-a + I b) = a^2 + b^2.

Hope this helps.

Raul Martinez





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