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Ein Function
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 of a function, 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 throug ... [...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) \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 a^\qquad\texta > 0,\ a \ne 1 Integrals involving the error function ... [...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 = 254.5\,\text long and the other L_2 = 253.5\,\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 = 255\,\text and \tilde L_2 = 253\,\text. These may be good approximations, in relative error, to the true lengths: the approximations are in error by less than 2% of the true lengths, , L_1 - \tilde L_1, /, L_1, < 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, |
Groundwater
Groundwater is the water present beneath Earth's surface in rock and soil pore spaces and in the fractures of rock formations. About 30 percent of all readily available freshwater in the world is groundwater. A unit of rock or an unconsolidated deposit is called an aquifer when it can yield a usable quantity of water. The depth at which soil pore spaces or fractures and voids in rock become completely saturated with water is called the water table. Groundwater is recharged from the surface; it may discharge from the surface naturally at springs and seeps, and can form oases or wetlands. Groundwater is also often withdrawn for agricultural, municipal, and industrial use by constructing and operating extraction wells. The study of the distribution and movement of groundwater is hydrogeology, also called groundwater hydrology. Typically, groundwater is thought of as water flowing through shallow aquifers, but, in the technical sense, it can also contain soil moisture, perma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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