Power Series Expansion for Cosine Integral Function plus Logarithm
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Theorem
- $\ds \map \Ci x = -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac {x^{2 n} } {\paren {2 n} \times \paren {2 n}!}$
where:
- $\Ci$ denotes the cosine integral function
- $\gamma$ denotes the Euler-Mascheroni constant
- $x$ is a strictly positive real number.
Proof
\(\ds \map \Ci x\) | \(=\) | \(\ds -\gamma - \ln x + \int_0^x \frac {1 - \cos u} u \rd u\) | Characterization of Cosine Integral Function | |||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma - \ln x + \int_0^x \frac 1 u \paren {1 - \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {u^{2 n} } {\paren {2 n}!} } \rd u\) | Power Series Expansion for Cosine Function | |||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \int_0^x \frac {u^{2 n - 1} } {\paren {2 n}!} \rd u\) | Power Series is Termwise Integrable within Radius of Convergence | |||||||||||
\(\ds \) | \(=\) | \(\ds -\gamma - \ln x + \sum_{n \mathop = 1}^\infty \paren {-1}^{n + 1} \frac {u^{2 n} } {\paren {2 n} \times \paren {2 n}!}\) | Primitive of Power |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Cosine Integral $\ds \map \Ci x = \int_x^\infty \frac {\cos u} u \rd u$: $35.15$