Ramanujan-Soldner constant

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

Its value is approximately "μ" ≈ 1.451369234883381050283968485892027449493…

Since the logarithmic integral is defined by

:$mathrm\{li\}(x)\; =\; int\_0^x\; frac\{dt\}\{ln\; t\},$

we have

:$mathrm\{li\}(x);=;mathrm\{li\}(x)\; -\; mathrm\{li\}(mu)$

:$int\_0^x\; frac\{dt\}\{ln\; t\}\; =\; int\_0^x\; frac\{dt\}\{ln\; t\}\; -\; int\_0^\{mu\}\; frac\{dt\}\{ln\; t\}$

:$mathrm\{li\}(x)\; =\; int\_\{mu\}^x\; frac\{dt\}\{ln\; t\},$

thus easing calculation for positive integers. Also, since the exponential integral function satisfies the equation

:$mathrm\{li\}(x);=;mathrm\{Ei\}(ln\{x\})$,

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…

External links

*MathWorld | urlname=SoldnersConstant | title=Soldner's Constant