Ramanujan-Soldner constant
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In mathematics, the Ramanujan-Soldner constant is a mathematical constant defined as the unique positive zero of the logarithmic integral function.
Its value is approximately μ ≈ 1.451369234883381050283968485892027449493...
Since the logarithmic integral is defined by
- <math> \mathrm{li}(x) = \int_0^x \frac{dt}{\ln t}, </math>
we have
- <math> \mathrm{li}(x)\;=\;\mathrm{li}(x) - \mathrm{li}(\mu) </math>
- <math> \int_0^x \frac{dt}{\ln t} = \int_0^x \frac{dt}{\ln t} - \int_0^{\mu} \frac{dt}{\ln t} </math>
- <math> \mathrm{li}(x) = \int_{\mu}^x \frac{dt}{\ln t}, </math>
thus easing calculation for positive integers. Also, since the exponential integral function satisfies the equation
- <math> \mathrm{li}(x)\;=\;\mathrm{Ei}(\ln{x}) </math>,
the only positive zero of the exponential integral occurs at the natural logarithm of the Ramanujan-Soldner constant, whose value is approximately ln(μ) ≈ 0.372507410781366634461991866...
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