Editing Ramanujan–Soldner constant

{{one source |date=April 2024}}
[[File:Logarithmic Integral Function and Soldner Constant.png|thumb|right|350px|Ramanujan–Soldner constant as seen on 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'' ≈ 1.45136923488338105028396848589202744949303228… {{OEIS|A070769}}

Since the logarithmic integral is defined by

:<math> \mathrm{li}(x) = \int_0^x \frac{dt}{\ln t}, </math>

then using <math> \mathrm{li}(\mu) = 0, </math> we have

:<math> \mathrm{li}(x)\;=\;\mathrm{li}(x) - \mathrm{li}(\mu) = \int_0^x \frac{dt}{\ln t} - \int_0^{\mu} \frac{dt}{\ln t} = \int_{\mu}^x \frac{dt}{\ln t},</math>

thus easing calculation for numbers greater than ''μ''.  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}}

[[Category:Mathematical constants]]
[[Category:Srinivasa Ramanujan]]


{{Math-stub}}