# Thin airfoil & lifting line theory principal value integrals

Reference: Batchelor, Introduction to Fluid Dynamics after eqn (6.9.8)

Assuming[{Element[n, Integers], n > 0, Element[a, Reals], a > 0,
Element[t, Reals], t > 0},
Integrate[Cos[n*t]/(Cos[t] - Cos[a]), {t, 0, Pi}, PrincipalValue -> True]]


A "standard" definite integral Pi* Sin[n a]/Sin[a]. However, I can't get Mathematica to show equality between its result and this. And it is not an obvious listing In Gradshetyn either. Batchelor did prove his result in a footnote from recursion after an indefinte integration for n = 0

Log[Abs[Sin[(a + t)/2]/Sin[(a - t)/2]]]/Sin[a]


which Mathematica does do. Please include successful code for the definite integrals. I was going to try Mathematica extending the theory to oscillating lift but a prerequisite is that it can do the steady case easily and not recursively.

• I don't see any clear statement of what you are asking us to do to help you. Could you please edit the question to provide that? – m_goldberg Feb 10 at 16:24
• For n=0 Mathematica finds $I_0=i\pi \csc {a}$ so the real part is 0. For n=1, Mathematica finds $I_1=\pi - i\pi \cot {a}$, so the real part is $\pi$, as well as in Batchelor 's book. Then use the recurrent formula. – Alex Trounev Feb 10 at 19:48
• Please post your successful code as I am still have trouble with the definite integrals in Mathematica. – simon Feb 10 at 21:24

1) $$n=0$$

Integrate[1/(Cos[t] - Cos[a]), {t, 0, Pi}, PrincipalValue -> True]
(*ConditionalExpression[
I \[Pi] Csc[
a], (Re[ArcCos[Cos[a]]] < 0 || Re[ArcCos[Cos[a]]] > \[Pi] ||
ArcCos[Cos[a]] \[NotElement] Reals) && Sin[a/2] >= 0 &&
Cos[a/2] <= 0]*)


2)$$n=1$$

Integrate[Cos[t]/(Cos[t] - Cos[a]), {t, 0, Pi},
PrincipalValue -> True]
(*ConditionalExpression[\[Pi] - I \[Pi] Cot[a],
Cos[a/2] >=
0 && (Re[ArcCos[Cos[a]]] < 0 || Re[ArcCos[Cos[a]]] > \[Pi] ||
ArcCos[Cos[a]] \[NotElement] Reals) && Sin[a/2] <= 0]*)


We are interested in the real part of the resulting expressions: $$n=0, I_0=0$$ and $$n=1, I_1=\pi$$ Thus, the real part of the resulting expressions is the same as in the book. Then use the recurrent formula $$I_{n+1}+I_{n-1}=2I_n\cos{a}$$

• It took awhile to see how to read these code answers and ignore the inserted backslashes and Mathematica's conditions. They are simpler than what v 7 gave me and allow me to see its Re part was also zero. Ignore. Ia >= [Pi] it had I0 =Csc[a] (I*Pi + Log[2] - Log[-2 Cos[a/2]] + Log[Cos[a/2]]) – simon Feb 11 at 1:18