Logarithmic Integral Function and Soldner Constant

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, the Ramanujan–Soldner constant (also called the Soldner constant) is a

mathematical constant A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathem ...

defined as the unique positive

zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

of the

logarithmic integral function In mathematics, the logarithmic integral function or integral logarithm li(''x'') is a special function. It is relevant in problems of physics and has number theory, number theoretic significance. In particular, according to the prime number the ...

. It is named after

Srinivasa Ramanujan Srinivasa Ramanujan Aiyangar (22 December 188726 April 1920) was an Indian mathematician. Often regarded as one of the greatest mathematicians of all time, though he had almost no formal training in pure mathematics, he made substantial con ...

and Johann Georg von Soldner. Its value is approximately ''μ'' ≈ 1.45136923488338105028396848589202744949303228… Since the logarithmic integral is defined by :

\mathrm(x) = \int_0^x \frac,

then using

\mathrm(\mu) = 0,

we have :

\mathrm(x)\;=\;\mathrm(x) - \mathrm(\mu) = \int_0^x \frac - \int_0^ \frac = \int_^x \frac,

thus easing calculation for numbers greater than ''μ''. Also, since the

exponential integral In mathematics, the exponential integral Ei is a special function on the complex plane. It is defined as one particular definite integral of the ratio between an exponential function and its argument. Definitions For real non-zero values of&nb ...

function satisfies the equation :

\mathrm(x)\;=\;\mathrm(\ln),

the only positive zero of the exponential integral occurs at the

natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...

of the Ramanujan–Soldner constant, whose value is approximately ln(''μ'') ≈ 0.372507410781366634461991866…

External links

* Mathematical constants Srinivasa Ramanujan {{Math-stub