Search results
Results from the WOW.Com Content Network
One of the simplest definitions is: The exponential function is the unique differentiable function that equals its derivative, and takes the value 1 for the value 0 of its variable. This "conceptual" definition requires a uniqueness proof and an existence proof, but it allows an easy derivation of the main properties of the exponential function.
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
The function e (−1/x 2) is not analytic at x = 0: the Taylor series is identically 0, although the function is not. If f ( x ) is given by a convergent power series in an open disk centred at b in the complex plane (or an interval in the real line), it is said to be analytic in this region.
An exact 100(1 − α)% confidence interval for the rate parameter of an exponential distribution is given by: [13] ^, < < ^,, which is also equal to ¯, < < ¯,, where χ 2 p , v is the 100( p ) percentile of the chi squared distribution with v degrees of freedom , n is the number of observations and x-bar is the sample average.
m + n + r + 1, r > 0, then the rational function R m, n occupies (r + 1) 2. cells in the Padé table, from position (m, n) through position (m+r, n+r), inclusive. In other words, if the same rational function appears more than once in the table, that rational function occupies a square block of cells within the table.
Logarithmic functions, using both base 10 and base e; Trigonometric functions (some including hyperbolic trigonometry) Exponential functions and roots beyond the square root; Quick access to constants such as π and e; In addition, high-end scientific calculators generally include some or all of the following:
If exponentiation is considered as a multivalued function then the possible values of (−1 ⋅ −1) 1/2 are {1, −1}. The identity holds, but saying {1} = {(−1 ⋅ −1) 1/2 } is incorrect. The identity ( e x ) y = e xy holds for real numbers x and y , but assuming its truth for complex numbers leads to the following paradox , discovered ...
The exponential function is an E-function, in its case c n = 1 for all of the n. If λ is an algebraic number then the Bessel function J λ is an E-function. The sum or product of two E-functions is an E-function. In particular E-functions form a ring. If a is an algebraic number and f(x) is an E-function then f(ax) will be an E-function.