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[[File:Logarithmic integral.png|thumb|The [[logarithmic integral function]].]]


In [[mathematics]], the '''Ramanujan–Soldner constant''' (also called the '''Soldner constant''') is a [[mathematical constant]] defined as the unique positive [[root of a function|zero]] of the [[logarithmic integral function]]. It is named after [[Srinivasa Ramanujan]] and [[Johann Georg von Soldner]].


Its value is approximately ''μ'' ≈ 1.451369234883381050283968485892027449493… {{OEIS|A070769}}
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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… {{OEIS|A091723}}
 
==External links==
*{{MathWorld | urlname=SoldnersConstant | title=Soldner's Constant}}
 
{{DEFAULTSORT:Ramanujan-Soldner constant}}
[[Category:Mathematical constants]]
[[Category:Srinivasa Ramanujan]]

Latest revision as of 11:34, 11 January 2015


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