Definite Integral from 0 to Pi of a Squared minus 2 a b Cosine x plus b Squared
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Theorem
- $\ds \int_0^\pi \map \ln {a^2 - 2 a b \cos x + b^2} \rd x = \begin{cases}2 \pi \ln a & a \ge b > 0 \\ 2 \pi \ln b & b \ge a > 0\end{cases}$
Proof
Note that:
- $\paren {a - b}^2 \ge 0$
so by Square of Sum:
- $a^2 - 2 a b + b^2 \ge 0$
So:
- $a^2 + b^2 \ge 2 a b = \size {-2 a b}$
so we may apply Definite Integral from $0$ to $\pi$ of $\map \ln {a + b \cos x}$.
We then have:
\(\ds \int_0^\pi \map \ln {a^2 - 2 a b \cos x + b^2} \rd x\) | \(=\) | \(\ds \pi \map \ln {\frac {a^2 + b^2 + \sqrt {\paren {a^2 + b^2}^2 - \paren {2 a b}^2} } 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \pi \map \ln {\frac {a^2 + b^2 + \sqrt {a^4 + 2 a^2 b^2 + b^4 - 4 a^2 b^2} } 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \pi \map \ln {\frac {a^2 + b^2 + \sqrt {a^4 - 2 a^2 b^2 + b^4} } 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \pi \map \ln {\frac {a^2 + b^2 + \sqrt {\paren {a^2 - b^2}^2} } 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \pi \map \ln {\frac {a^2 + b^2 + \size {a^2 - b^2} } 2}\) | Definition of Absolute Value |
Note that if $a \ge b > 0$ we have:
\(\ds \pi \map \ln {\frac {a^2 + b^2 + \size {a^2 - b^2} } 2}\) | \(=\) | \(\ds \pi \map \ln {\frac {a^2 + b^2 + a^2 - b^2} 2}\) | since $a \ge b > 0$, we have $a^2 \ge b^2$ and $\size {a^2 - b^2} = a^2 - b^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \pi \map \ln {a^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \pi \ln a\) | Logarithm of Power |
Note that if $b \ge a > 0$ we have:
\(\ds \pi \map \ln {\frac {a^2 + b^2 + \size {a^2 - b^2} } 2}\) | \(=\) | \(\ds \pi \map \ln {\frac {a^2 + b^2 + b^2 - a^2} 2}\) | since $a \ge b > 0$, we have $a^2 \ge b^2$ and $\size {a^2 - b^2} = b^2 - a^2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \pi \map \ln {b^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \pi \ln b\) | Logarithm of Power |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Logarithmic Functions: $15.108$