Search results
Results from the WOW.Com Content Network
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 theoretic significance. In particular, according to the prime number theorem , it is a very good approximation to the prime-counting function , which is defined as the number of prime numbers ...
The natural logarithm of t can be defined as the definite integral: ln t = ∫ 1 t 1 x d x . {\displaystyle \ln t=\int _{1}^{t}{\frac {1}{x}}\,dx.} This definition has the advantage that it does not rely on the exponential function or any trigonometric functions; the definition is in terms of an integral of a simple reciprocal.
The following is a list of integrals (antiderivative functions) of logarithmic functions. For a complete list of integral functions, see list of integrals. Note: x > 0 is assumed throughout this article, and the constant of integration is omitted for simplicity.
If the natural logarithm is defined as the integral =, then the derivative immediately follows from the first part of the fundamental theorem of calculus. On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative (for x > 0 ) can be found by using the properties of the ...
The complex logarithm is needed to define exponentiation in which the base is a complex number. Namely, if a {\displaystyle a} and b {\displaystyle b} are complex numbers with a ≠ 0 {\displaystyle a\not =0} , one can use the principal value to define a b = e b Log a {\displaystyle a^{b}=e^{b\operatorname {Log} a}} .
Note that the subtraction identity is not defined if =, since the logarithm of zero is not defined. Also note that, when programming, a {\displaystyle a} and c {\displaystyle c} may have to be switched on the right hand side of the equations if c ≫ a {\displaystyle c\gg a} to avoid losing the "1 +" due to rounding errors.
The binary logarithm function may be defined as the inverse function to the power of two function, which is a strictly increasing function over the positive real numbers and therefore has a unique inverse. [7] Alternatively, it may be defined as ln n/ln 2, where ln is the natural logarithm, defined in any of its
The Darboux integral, which is defined by Darboux sums (restricted Riemann sums) yet is equivalent to the Riemann integral. A function is Darboux-integrable if and only if it is Riemann-integrable. Darboux integrals have the advantage of being easier to define than Riemann integrals.