Logarithm of One plus x in terms of Gaussian Hypergeometric Function
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Theorem
- $\map \ln {1 + x} = x \map F {1, 1; 2; -x}$
where:
- $x$ is a real number with $\size x < 1$
- $F$ denotes the Gaussian hypergeometric function.
Proof
\(\ds x \map F {1, 1; 2; -x}\) | \(=\) | \(\ds x \sum_{n \mathop = 0}^\infty \frac {\paren {1^{\overline n} }^2} {2^{\overline n} } \frac {\paren {-x}^n} {n!}\) | Definition of Gaussian Hypergeometric Function | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \paren {n!}^2 \paren {\frac {\paren {2 - 1}!} {\paren {2 + n - 1}!} } \frac {x^{n + 1} } {n!}\) | Rising Factorial as Quotient of Factorials, One to Integer Rising is Integer Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {n!}^2} {\paren {n + 1}! n!} x^{n + 1}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {\paren {n!}^2} {\paren {n + 1} \paren {n!}^2} x^{n + 1}\) | Definition 1 of Factorial | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \paren {-1}^{n - 1} \frac {x^n} n\) | shifting the index | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \ln {1 + x}\) | Power Series Expansion for $\map \ln {1 + x}$ |
$\blacksquare$
Also presented as
Some sources give this as:
- $\map F {1, 1; 2; -x} = \dfrac {\map \ln {1 + x} } x$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 31$: Hypergeometric Functions: Special Cases: $31.4$