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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 ''x'', the exponential integral Ei(''x'') is defined as : \operatorname(x) = -\int_^\infty \fract\,dt = \int_^x \fract\,dt. The Risch algorithm shows that Ei is not an elementary function. The definition above can be used for positive values of ''x'', but the integral has to be understood in terms of the Cauchy principal value due to the singularity of the integrand at zero. For complex values of the argument, the definition becomes ambiguous due to branch points at 0 and Instead of Ei, the following notation is used, :E_1(z) = \int_z^\infty \frac\, dt,\qquad, (z), 0. Properties Several properties of the exponential integral below, in certain cases, allow one to avoid its explicit evaluation through the definition ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Integrals Of Exponential Functions
The following is a list of integrals of exponential functions. For a complete list of integral functions, please see the list of integrals. Indefinite integral Indefinite integrals are antiderivative functions. A constant (the constant of integration) may be added to the right hand side of any of these formulas, but has been suppressed here in the interest of brevity. Integrals of polynomials * \int xe^\,dx = e^\left(\frac\right) \qquad \text c \neq 0; * \int x^2 e^\,dx = e^\left(\frac-\frac+\frac\right) * \begin \int x^n e^\,dx &= \frac x^n e^ - \frac\int x^ e^ \,dx \\ &= \left( \frac \right)^n \frac \\ &= e^\sum_^n (-1)^i\fracx^ \\ &= e^\sum_^n (-1)^\fracx^i \end * \int\frac\,dx = \ln, x, +\sum_^\infty\frac * \int\frac\,dx = \frac\left(-\frac+c\int\frac\,dx\right) \qquad\textn\neq 1\text Integrals involving only exponential functions * \int f'(x)e^\,dx = e^ * \int e^\,dx = \frac e^ * \int a^\,dx = \frac\qquad\texta > 0,\ a \ne 1 Integrals involving the er ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catastrophic Cancellation
In numerical analysis, catastrophic cancellation is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers. For example, if there are two studs, one L_1 = 253.51\,\text long and the other L_2 = 252.49\,\text long, and they are measured with a ruler that is good only to the centimeter, then the approximations could come out to be \tilde L_1 = 254\,\text and \tilde L_2 = 252\,\text. These may be good approximations, in relative error, to the true lengths: the approximations are in error by less than 0.2% of the true lengths, , L_1 - \tilde L_1, /, L_1, < 0.2\%. However, if the ''approximate'' lengths are subtracted, the difference will be , even though the true difference between the lengths is . The difference of the approximations, |
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'' ≈ 1.45136923488338105028396848589202744949303228… 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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Power Series
In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a constant called the ''center'' of the series. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function. In many situations, the center ''c'' is equal to zero, for instance for Maclaurin series. In such cases, the power series takes the simpler form \sum_^\infty a_n x^n = a_0 + a_1 x + a_2 x^2 + \dots. The partial sums of a power series are polynomials, the partial sums of the Taylor series of an analytic function are a sequence of converging polynomial approximations to the function at the center, and a converging power series can be seen as a kind of generalized polynom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Function
In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon X\to Y, its inverse f^\colon Y\to X admits an explicit description: it sends each element y\in Y to the unique element x\in X such that . As an example, consider the real-valued function of a real variable given by . One can think of as the function which multiplies its input by 5 then subtracts 7 from the result. To undo this, one adds 7 to the input, then divides the result by 5. Therefore, the inverse of is the function f^\colon \R\to\R defined by f^(y) = \frac . Definitions Let be a function whose domain is the set , and whose codomain is the set . Then is ''invertible'' if there exists a function from to such that g(f(x))=x for all x\in X and f(g(y))=y for all y\in Y. If is invertible, then there is exactly one functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometric Integral
In mathematics, trigonometric integrals are a indexed family, family of nonelementary integrals involving trigonometric functions. Sine integral The different sine integral definitions are \operatorname(x) = \int_0^x\frac\,dt \operatorname(x) = -\int_x^\infty\frac\,dt~. Note that the integrand \frac is the sinc function, and also the zeroth Bessel function#Spherical Bessel functions: jn.2C yn, spherical Bessel function. Since is an even function, even entire function (holomorphic over the entire complex plane), is entire, odd, and the integral in its definition can be taken along Cauchy's integral theorem, any path connecting the endpoints. By definition, is the antiderivative of whose value is zero at , and is the antiderivative whose value is zero at . Their difference is given by the Dirichlet integral, \operatorname(x) - \operatorname(x) = \int_0^\infty\frac\,dt = \frac \quad \text \quad \operatorname(x) = \frac + \operatorname(x) ~. In signal processing, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Incomplete Gamma Function
In mathematics, the upper and lower incomplete gamma functions are types of special functions which arise as solutions to various mathematical problems such as certain integrals. Their respective names stem from their integral definitions, which are defined similarly to the gamma function but with different or "incomplete" integral limits. The gamma function is defined as an integral from zero to infinity. This contrasts with the lower incomplete gamma function, which is defined as an integral from zero to a variable upper limit. Similarly, the upper incomplete gamma function is defined as an integral from a variable lower limit to infinity. Definition The upper incomplete gamma function is defined as: \Gamma(s,x) = \int_x^ t^\,e^\, dt , whereas the lower incomplete gamma function is defined as: \gamma(s,x) = \int_0^x t^\,e^\, dt . In both cases is a complex parameter, such that the real part of is positive. Properties By integration by parts we find the recurrence relati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 theorem, it is a very good approximation to the prime-counting function, which is defined as the number of prime numbers less than or equal to a given value . Integral representation The logarithmic integral has an integral representation defined for all positive real numbers ≠ 1 by the integral, definite integral : \operatorname(x) = \int_0^x \frac. Here, denotes the natural logarithm. The function has a mathematical singularity, singularity at , and the integral for is interpreted as a Cauchy principal value, : \operatorname(x) = \lim_ \left( \int_0^ \frac + \int_^x \frac \right). Offset logarithmic integral The offset logarithmic integral or Eulerian logarithmic integral is defined as : \operatorname(x) = \int_2^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Confluent Hypergeometric Functions
In mathematics, a confluent hypergeometric function is a solution of a confluent hypergeometric equation, which is a degenerate form of a hypergeometric differential equation where two of the three regular singularities merge into an irregular singularity. The term ''confluent'' refers to the merging of singular points of families of differential equations; ''confluere'' is Latin for "to flow together". There are several common standard forms of confluent hypergeometric functions: * Kummer's (confluent hypergeometric) function , introduced by , is a solution to Kummer's differential equation. This is also known as the confluent hypergeometric function of the first kind. There is a different and unrelated Kummer's function bearing the same name. * Tricomi's (confluent hypergeometric) function introduced by , sometimes denoted by , is another solution to Kummer's equation. This is also known as the confluent hypergeometric function of the second kind. * Whittaker functions (for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Numbers
In mathematics, the -th harmonic number is the sum of the Multiplicative inverse, reciprocals of the first natural numbers: H_n= 1+\frac+\frac+\cdots+\frac =\sum_^n \frac. Starting from , the sequence of harmonic numbers begins: 1, \frac, \frac, \frac, \frac, \dots Harmonic numbers are related to the harmonic mean in that the -th harmonic number is also times the reciprocal of the harmonic mean of the first positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series (mathematics), harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions. The harmonic numbers roughly approximate the natural logarithm, natural logarithm function and thus the associated harmonic series (mathematics), harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the Divergence of the sum of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entire Function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function f(z) has a root at w, then f(z)/(z-w), taking the limit value at w, is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function. A transcendental entire function is an entire function that is not a polynomial. Just as meromorphic functions can be viewed as a generalization of rational fractions, entire functions can be viewed as a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
