[Date Index]
[Thread Index]
[Author Index]
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
Prev by Date:
**Re: proof of formula for log(-t) found in Mathematica?**
Next by Date:
**Re: Precision of output**
Previous by thread:
**proof of formula for log(-t) found in Mathematica?**
Next by thread:
**Re: proof of formula for log(-t) found in Mathematica?**
| |