|
List Of Laplace Transforms
The following is a list of Laplace transforms for many common functions of a single variable. The Laplace transform is an integral transform that takes a function of a positive real variable (often time) to a function of a complex variable (complex angular frequency). Properties The Laplace transform of a function f(t) can be obtained using the formal definition of the Laplace transform. However, some properties of the Laplace transform can be used to obtain the Laplace transform of some functions more easily. Linearity For functions f and g and for scalar a, the Laplace transform satisfies : \mathcal\ = a \mathcal\ + b \mathcal\ and is, therefore, regarded as a linear operator. Time shifting The Laplace transform of f(t - a) u(t - a) is e^ F(s) . Frequency shifting The Laplace transform of e^ f(t) is F(s - a) . Explanatory notes The unilateral Laplace transform takes as input a function whose time domain is the non-negative reals, which is why all of the ti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Transform
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a function of a Complex number, complex variable s (in the complex-valued frequency domain, also known as ''s''-domain, or ''s''-plane). The transform is useful for converting derivative, differentiation and integral, integration in the time domain into much easier multiplication and Division (mathematics), division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic equation, algebraic polynomial equations, and by simplifyin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Fourier Transforms
In mathematics, the Fourier transform (FT) is an integral transform that takes a function (mathematics), function as input then outputs another function that describes the extent to which various Frequency, frequencies are present in the original function. The output of the transform is a complex number, complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the Operation (mathematics), mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical Chord (music), chord into the sound intensity, intensities of its constituent Pitch (music), pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the #Uncertainty principle, uncerta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
