What to do when FullSimplify runs out of memory?

General thoughts on simplifying very large expressions. Obviously, these are not applicable in all cases.

• Simplify the parts before assembling the large expression.
• Identify common subexpressions. See if these can be simplified individually.
• Try less expensive simplifications (e.g. Refine, Simplify, Factor, rather than FullSimplify)
• Apply assumptions, where relevant, using Refine to apply them.
• Try Map[Simplify, expression,{n}] to try simplifying subexpressions at level n, rather than trying to simplify their combined form.
• Use LeafCount to assess whether you are reducing the size at each step.
• Substitute numerical values to assess whether your assumptions (e.g. that the expression is real valued) are correct.
• In general, if simplifications do not succeed rapidly, they are rarely going to be useful.
• Use ComplexExpand (carefully) where your variables are real, but the expression is complex.
• See whether a substituting a numerical value (e.g 0 for additive constants, 1 for multiplicative constants) for some of your parameters allows a simplified form to be found. In your example, you might try a->1. If not, you are unlikely to succeed symbolically.

As a trivial example, I show how I might tackle simplifying all the Sinand Cos terms in an expression.

example = Sin[x^2 + 2 x + 1] + Sin[x^3 + 3 x^2 + 3 x + 1] Sin[x^2 + 2 x + 1] (* Sin[1 + 2 x + x^2] + Sin[1 + 2 x + x^2] Sin[1 + 3 x + 3 x^2 + x^3] *) Cases[example, _Sin | _Cos, ∞] // Union (* {Sin[1 + 2 x + x^2], Sin[1 + 3 x + 3 x^2 + x^3]} *) Simplify[%] (* {Sin[(1 + x)^2], Sin[(1 + x)^3]} *) example /. Thread[%% -> %] (* Sin[(1 + x)^2] + Sin[(1 + x)^2] Sin[(1 + x)^3] *) LeafCount /@ {example, %} (* {34, 20} *) 

In the Wolfram Language, FullSimplify is a powerful function used to simplify complex expressions, but it may encounter difficulties in simplifying expressions involving exponential functions (Exp) and complex numbers. Here are some common problems you might encounter and some strategies to address them:

1. Simplification with Complex Numbers:

   FullSimplify[Exp[I x], x ∈ Reals] 

   In this case, you might expect the result to simplify to Cos[x] + I Sin[x]. However, FullSimplify might not always recognize this simplification, especially if the assumption about x being real is not explicitly provided.

   To address this, you can use assumptions or provide additional transformation rules:

   FullSimplify[Exp[I x], x ∈ Reals, TransformationFunctions -> {Automatic, TrigReduce}] 

2. Simplification involving Exponential Functions:

   FullSimplify[Exp[x] Exp[y], x > 0 && y > 0] 

   In this case, you might expect the result to simplify to Exp[x + y]. However, FullSimplify may not always perform this simplification automatically.

   One approach is to use transformation rules explicitly:

   FullSimplify[Exp[x] Exp[y], x > 0 && y > 0, TransformationFunctions -> {Automatic, # /. Exp[a_] Exp[b_] :> Exp[a + b] &}] 

3. Handling Complex Conjugates:

   FullSimplify[Conjugate[Exp[I x]], x ∈ Reals] 

   You might expect the result to simplify to Exp[-I x], but FullSimplify may not recognize this simplification directly.

   One way to handle this is to use ComplexExpand before simplification:

   FullSimplify[ComplexExpand[Conjugate[Exp[I x]]], x ∈ Reals] 

4. Involving Special Functions: FullSimplify may struggle with simplifications involving special functions like Gamma, Beta, etc., especially when combined with exponentials and complex numbers.

   In such cases, it's helpful to provide specific assumptions and use appropriate transformation functions tailored to your problem domain.

Overall, when dealing with complex expressions involving exponentials and complex numbers in the Wolfram Language, it's essential to carefully set assumptions and use transformation functions to guide FullSimplify to the desired results. Additionally, sometimes a combination of manual simplifications and transformation rules might be necessary to achieve the desired simplification.