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

**References**:**proof of formula for log(-t) found in Mathematica?***From:*"Roger L. Bagula" <rlbtftn@netscape.net>