I have the following function (it is the incomplete elliptic integral of first kind) $$ F(b,g) = \int_{0}^{b} \frac{dx}{\sqrt{(1-x^2)(1-gx^2)}} $$ I would like to compute $$\frac{\partial F}{\partial g} \ ,\ \frac{\partial F}{\partial b} \ ,\ \frac{\partial^2 F}{\partial g^2} \ ,\ \frac{\partial^2 F}{\partial b^2} \ ,\ \frac{\partial^2 F}{\partial b\partial g}$$ so I defined

F[b_,g_]:= Integrate[1/Sqrt[(1 - x^2)*(1 - g*x^2)], {x, 0, b}]

and tried the command


but Mathematica cannot compute it. Am I doing something wrong or is there a way to do it?

  • 1
    $\begingroup$ Why not use EllipticF instead of an integal? E.g., F[b_, g_] := EllipticF[ArcSin[b], g]. $\endgroup$
    – Carl Woll
    Oct 20, 2020 at 16:26
  • $\begingroup$ An indirect way: try Integrate[f[x, g], {x, 0, b}] // D[#, g] & , then replace f with real func. $\endgroup$ Oct 20, 2020 at 16:31
  • $\begingroup$ @CarlWoll because I want to keep it as general as possible in order to modify it easily someday $\endgroup$ Oct 20, 2020 at 16:36
  • $\begingroup$ Derivatives relative to "b" are trivial and why not swap the derivative relative to g and the integral? $\endgroup$ Oct 20, 2020 at 16:46

1 Answer 1


I don't know if you want to be this restrictive, but you can avoid singularities that Mathematica evidently has trouble with by making assumptions.

F[b_,g_]:= Integrate[1/Sqrt[(1 - x^2)*(1 - g*x^2)], {x, 0, b}]

$Assumptions = 0 < b < 1 && 0 < g < 1

D[F[b, g], g]

$\frac{\sin \left(2 \sin ^{-1}(b)\right)}{4 (g-1) \sqrt{1-b^2 g}}-\frac{F\left(\left.\sin ^{-1}(b)\right|g\right)}{2 g}-\frac{E\left(\left.\sin ^{-1}(b)\right|g\right)}{2 (g-1) g}$

The other derivatives work also.


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