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Since e is an irrational number (see proof that e is irrational), it cannot be represented as the quotient of two integers, but it can be represented as a continued fraction. Using calculus , e may also be represented as an infinite series , infinite product , or other types of limit of a sequence .
r = | z | = √ x 2 + y 2 is the magnitude of z and; φ = arg z = atan2(y, x). φ is the argument of z, i.e., the angle between the x axis and the vector z measured counterclockwise in radians, which is defined up to addition of 2π. Many texts write φ = tan −1 y / x instead of φ = atan2(y, x), but the first equation needs ...
Following a proposal by William Kahan, it may thus be useful to have a dedicated routine, often called expm1, which computes e x − 1 directly, bypassing computation of e x. For example, one may use the Taylor series: e x − 1 = x + x 2 2 + x 3 6 + ⋯ + x n n ! + ⋯ . {\displaystyle e^{x}-1=x+{\frac {x^{2}}{2}}+{\frac {x^{3}}{6}}+\cdots ...
Definition (3) presents a problem because there are non-equivalent paths along which one could integrate; but the equation of (3) should hold for any such path modulo . As for definition (5), the additive property together with the complex derivative f ′ ( 0 ) = 1 {\displaystyle f'(0)=1} are sufficient to guarantee f ( x ) = e x ...
The number e (e = 2.71828...), also known as Euler's number, which occurs widely in mathematical analysis The number i , the imaginary unit such that i 2 = − 1 {\displaystyle i^{2}=-1} The equation is often given in the form of an expression set equal to zero, which is common practice in several areas of mathematics.
One of the most important forms in which an analytic continued fraction can appear is as a regular continued fraction, which is a continued fraction of the form f ( z ) = b 0 + a 1 z 1 − a 2 z 1 − a 3 z 1 − a 4 z 1 − ⋱ . {\displaystyle f(z)=b_{0}+{\cfrac {a_{1}z}{1-{\cfrac {a_{2}z}{1-{\cfrac {a_{3}z}{1-{\cfrac {a_{4}z}{1-\ddots }}}}}}}}.}
Exponential functions y = 2 x and y = 4 x intersect the graph of y = x + 1, respectively, at x = 1 and x = -1/2. The number e is the unique base such that y = e x intersects only at x = 0. We may infer that e lies between 2 and 4. The number e is the unique real number such that (+) < < (+) + for all positive x. [31]
ρ can be computed from the series expansion: [10] = = ()! (+) For rational values of β , ρ ( u ) can be calculated in terms of elementary functions. But the expression is in general too complex to be useful except for the case β = 1/2 where G ( u ) = u ρ ( u ) = 1 2 π u e − u / 4 {\displaystyle G(u)=u\rho (u)={1 \over 2{\sqrt {\pi ...