Dirichlet Beta Function at Odd Positive Integers/Examples/Dirichlet Beta Function of 1
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Theorem
Let $\beta$ denote the Dirichlet beta function.
Then:
- $\map \beta 1 = \dfrac \pi 4 $
Proof 1
\(\ds \map \beta {2 n + 1}\) | \(=\) | \(\ds \paren {-1}^n \dfrac {E_{2 n} \pi^{2 n + 1} } {4^{n + 1} \paren {2 n}!}\) | Dirichlet Beta Function at Odd Positive Integers | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \beta 1\) | \(=\) | \(\ds \paren {-1}^0 \dfrac {E_0 \pi} {4 \paren 0!}\) | setting $n \gets 0$ | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac \pi 4\) | Factorial of Zero, Zeroth Power of Real Number equals One and Euler Number Values:$E_0 = 1$ |
$\blacksquare$
Proof 2
\(\ds \frac 1 {1 + x^2}\) | \(=\) | \(\ds 1 - x^2 + x^4 - x^6 + \cdots\) | Sum of Infinite Geometric Sequence | |||||||||||
\(\ds \int_0^1 \frac 1 {1 + x^2} \rd x\) | \(=\) | \(\ds \int_0^1 \paren {1 - x^2 + x^4 - x^6 + \cdots } \rd x\) | integrating both sides from $0$ to $1$ | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \map \arctan 1 - \map \arctan 0\) | \(=\) | \(\ds \intlimits {x - \frac {x^3} 3 + \frac {x^5} 5 - \frac {x^7} 7 + \cdots } 0 1\) | Derivative of Arctangent Function, Primitive of Power | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac \pi 4 - 0\) | \(=\) | \(\ds 1 - \frac 1 3 + \frac 1 5 - \frac 1 7 + \cdots\) | Arctangent of One, Arctangent of Zero is Zero | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac \pi 4\) | \(=\) | \(\ds \map \beta 1\) | Definition of Dirichlet Beta Function |
$\blacksquare$