# Elliptic Integral simplification

Integrate[Sqrt[(roh^2 - r^2)/(r^2 - rb^2)], {r, rb, roh},
Assumptions -> r > rb \[And] roh > rb \[And] roh > 1]


Outputs to:

 ConditionalExpression[(
rb roh (-Sqrt[(1/rb)] Sqrt[-rb] EllipticE[rb^2/roh^2] +
I EllipticE[ArcSin[roh/rb], rb^2/roh^2]))/Abs[rb], rb + roh >= 0]


EDIT1:

Integrate[Sqrt[(roh^2 - r^2)/(r^2 - rb^2)], {r, rb, roh},
Assumptions -> rb > 0 && roh > rb]
Integrate[Sqrt[(m^2 - r^2)/(r^2 - m^2 + 1)], {r, Sqrt[m^2 - 1], m},
Assumptions -> m > 1]
Clear["Global*"];
Plot[-I m (EllipticE[1 - 1/m^2] -
EllipticE[ArcSin[m/Sqrt[-1 + m^2]], 1 - 1/m^2]), {m, 1 + 10^-6,
4}, GridLines -> Automatic]


Additionally to simplify using a single parameter $$m>1$$ defined through $$(roh = m, rb= \sqrt{m^2-1})$$ one obtains plot:

Thanks for understanding/comment/intuition /about how a plot with $$\sqrt{-1}$$ coefficient can work, when arguments of $$(K,E)$$ type elliptic integrals appear to be real.

WA plot

• Looking at result, without assumptions shows a term $\sqrt\frac{-1}{\rho}$. But then you say in the assumptions that $\rho>1$, therefore the complex number shows up. To get rid of the complex number, then remove the assumption that causes it. Sep 4 '21 at 1:04
• If an expression with complex components reduce to real than it must be the imaginary portions cancel.
– josh
Dec 11 '21 at 16:43

Integrate[Sqrt[(roh^2 - r^2)/(r^2 - rb^2)], {r, rb, roh}, Assumptions -> rb > 0 && roh > rb ]
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