When I use the Mathematica to obtain the integral of the symbolic expression, the output result seems not to be applied for all the cases. For example I use the code
Integrate[Cos[m Pi x] Cos[n Pi x], {x, 0, 1}, Assumptions -> m \[Element] Integers && n \[Element] Integers] to calculate the integral $$\int\cos (m\pi x)\cos (n \pi x)\,\mathrm{d}x$$
The output is $$\frac{m \sin (\pi m) \cos (\pi n)-n \cos (\pi m) \sin (\pi n)}{\pi m^2-\pi n^2}$$
I define a function with respect to the variables $m$ and $n$
f[m_, n_] := ( m Cos[n \[Pi]] Sin[m \[Pi]] - n Cos[m \[Pi]] Sin[n \[Pi]])/( m^2 \[Pi] - n^2 \[Pi]); Obviously, the function $f(m,n)$ is only applicable for the cases $m\neq n$. Some assumptions are made during the calculation by Mathematica. Now I want to obatin the unified expressions for the integral, for example, The result can be a conditional expression.
f(m,n)is applicable for the casesm=n, if one takes the limitn->minstead of assignment. Try enhance the definition:f[m_, m_]:=Limit[f[m, n], n -> m]. $\endgroup$Limitis very time-consuming. Are there any alternative methods? $\endgroup$