# Assumptions with Multiple Rules

What is the correct way to add more than one assumption while integrating?

I am attempting to evaluate the following integral and Mathematica is just stalling, so I'm wondering if my assumptions are the problem:

 Integrate[k Sqrt[k^2 - k0^2] ((kf^2 - k^2)/k^2 Log[(kf + k)/(kf - k)]
+ 2 kf), {k, k0, Sqrt[kf^2 + k0^2] - e}, Assumptions -> {Reals, e > 0}]

• What do you mean by Reals in the Assumptions? That all variables appearing in the integrand are real? or that only e is real? – QuantumDot May 11 '16 at 2:11
• e must be Real anyway since it is greater than 0 – censored user May 11 '16 at 2:44
• @QuantumDot Since Assumptions->Reals means all variables involved are real, I was trying to retain that meaning while also adding the e>0 condition. – Max May 11 '16 at 3:52
• Possible duplicate of Usage of Assuming for Integration – user9660 May 11 '16 at 4:06
• To be safe you might try to be explicit about the real variables: Assumptions -> {k0, kf, ky} \[Element] Reals && e > 0 – Rashid May 11 '16 at 4:16

I think what you mean is

Assuming[a<0 && b\[Element]Reals && c==3, FullSimplify[Integrate[f[a,b,c,d], {d,e,f}]]]


if you have different assumptions for different variables, or with the same assumption for a bunch of variables:

Assuming[{a, b, c}>0 && a>b, FullSimplify[Integrate[f[a,b,c,d], {d,e,f}]]]


so use the Assuming[] and && commands.

The difference between the Assuming[] and the Assumptions-> command is the topic of this thread: click

• Two question about this: Is there a way to specify the assumption that all variables involved are real using this method, the way Assumptions->Reals does? And I thought I had read that using Assuming[a==1, ...] permanently added "a==1" to the default assumptions for further calculations until changed, so I had avoided it. Is that true or am I mistaken? – Max May 11 '16 at 3:57
• @Max, Assuming only temporarily appends to $Assumptions for that expression -- it won't affect further calculations. To see this, try comparing $Assumptions with Assuming[x > 0, \$Assumptions] ...That said, I usually do what you did by adding an assumption to Integrate. I find that more readable. – Rashid May 11 '16 at 4:11
• Rashid already answered that, but for your integral there seems to be no analytical solution with the given assumptions, so you might have to use NIntegrate instead of Integrate. – censored user May 11 '16 at 4:23