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Substitution of Exp() and avoiding division by zero in simplification?

Posted 8 years ago

I have a complicated expression that I'm trying to set to zero and solve (expression Eq at the bottom). I simplify by substituting in variables, but Mathematica isn't fully simplifying it -- my guess is because it thinks it might be dividing by zero. I try to avoid this, but I'm getting stuck. Would be very appreciative of help with this and with ultimately solving the equation for b.

When I try solving directly, Mathematica just returns the following uncomputed:

Solve[Eq == 0, b]

I know Eq can be further simplified by canceling out the (A-[Mu]) term from the numerator and the denominators, but this isn't happening. Is there a way I can tell Mathematica that A != [Mu] ? Also, there are some additional conditions (in addition to everything being real) on the variables that hopefully make it easier for Mathematica to solve:

A > 0 && B > 0 && S > 0 && b > 0 && \[Lambda] > 0 && \[Sigma] > 0 && me > 0 && mc > 0 && \[Gamma] > 0 && A != \[Mu], Reals

Also, to further simplify I'm trying to replace all the occurences of e^b with a variable Y in the expression Eq (full expression below) but I'm striking out. In addition to attempts at using Replace and Eliminate, he following two don't work:

Simplify[Eq /.Exp[b] -> Y]
Simplify[Eq /.E^b -> Y]

I've already made the following substitutions:

    A = Sqrt[2 \[Lambda] \[Sigma]^2+\[Mu]^2]
    B = \[Mu]/\[Sigma]^2
    S = \[Sigma]^2

The expression I'd like to set to zero and solve for b:

Eq = \[Gamma]+(2 A E^((3 A b)/S) (E^((A b)/S) mc+E^(b B) me) ((E^((A b)/(2 S)) Sqrt[E^((A b)/S) mc+E^(b B) me])/Sqrt[mc+E^((b (A+\[Mu]))/S) me])^(-((A+\[Mu])/A)) (-2 S \[Lambda]+(A-\[Mu]) \[Mu]))/((-1+E^((2 A b)/S))^2 S \[Lambda] (A-\[Mu]))-(A E^((A b)/S) (E^((A b)/S) mc+E^(b B) me) ((E^((A b)/(2 S)) Sqrt[E^((A b)/S) mc+E^(b B) me])/Sqrt[mc+E^((b (A+\[Mu]))/S) me])^(-((A+\[Mu])/A)) (-2 S \[Lambda]+(A-\[Mu]) \[Mu]))/((-1+E^((2 A b)/S)) S \[Lambda] (A-\[Mu]))-(E^((A b)/S) ((E^((A b)/(2 S)) Sqrt[E^((A b)/S) mc+E^(b B) me])/Sqrt[mc+E^((b (A+\[Mu]))/S) me])^(-((A+\[Mu])/A)) (A E^((A b)/S) mc+E^(b B) me \[Mu]) (-2 S \[Lambda]+(A-\[Mu]) \[Mu]))/((-1+E^((2 A b)/S)) S \[Lambda] (A-\[Mu]))+(mc ((E^((A b)/(2 S)) Sqrt[E^((A b)/S) mc+E^(b B) me])/Sqrt[mc+E^((b (A+\[Mu]))/S) me])^(1-\[Mu]/A) (A+\[Mu]) (-2 S \[Lambda]+(A-\[Mu]) \[Mu]) (2 A E^((A b)/S) mc+E^((b (2 A+\[Mu]))/S) me (A-\[Mu])+E^(b B) me (A+\[Mu])))/(2 A (-1+E^((2 A b)/S)) (E^((A b)/S) mc+E^(b B) me) S \[Lambda] (A-\[Mu]))
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