Assumptions with Multiple Rules

Question

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
@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
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

Community · Accepted Answer · 2017-04-13 12:55:40Z

2

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

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answered May 11 '16 at 2:49

censored user

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$\begingroup$ 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? $\endgroup$ – Max May 11 '16 at 3:57
$\begingroup$ @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. $\endgroup$ – Rashid May 11 '16 at 4:11
$\begingroup$ 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. $\endgroup$ – censored user May 11 '16 at 4:23

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