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Re: How to solve this equation?

• To: mathgroup at smc.vnet.net
• Subject: [mg103533] Re: [mg103518] How to solve this equation?
• From: danl at wolfram.com
• Date: Sun, 27 Sep 2009 07:30:46 -0400 (EDT)
• References: <200909261013.GAA23377@smc.vnet.net>

> Hi,
>
> Can anyone tell me how to solve a^{-x}+b^{-x}+c^{-x}==0 for x, where a and
> b and c are some constants.
>
> Yours,
>
> Jamil.

For given numeric constants, could use FindRoot to get one solution.

In[123]:= FindRoot[Pi^(-x) + E^(-x) + Catalan^(-x) == 0, {x, .1 + I}]
Out[123]= {x -> 0.581515 + 2.71516 I}

Or could use reduce, to get all roots in some region.

In[12]:= Reduce[Pi^(-x) + E^(-x) + Catalan^(-x) == 0 && Abs[x] < 5, x]
Out[12]= x ==
  Root[{1 + E^(-(1 - Log[\[Pi]]) #1) +
      E^(-(Log[Catalan] - Log[\[Pi]]) #1) &,
    0.581514699561345300020023543344396616265085370855763 -
     2.715155022486549482573351321493738583224908371244565 I}] ||
 x == Root[{1 + E^(-(1 - Log[\[Pi]]) #1) +
      E^(-(Log[Catalan] - Log[\[Pi]]) #1) &,
    0.581514699561345300020023543344397 +
     2.715155022486549482573351321493739 I}]

This result can be viewed as local root-finding plus validation, meaning
it gives all roots in the specified region. A caveat is the region must
only contain finitely many roots.

Daniel Lichtblau
Wolfram Research

• References:
  • How to solve this equation?
    • From: Jamil Ariai <j_ariai@hotmail.com>