Top 10 similar words or synonyms for weierstrass_transform
laplace_transform 0.802383
laplacian 0.800328
dirac_delta_function 0.798979
hilbert_transform 0.797611
poisson_kernel 0.793095
frobenius_norm 0.792065
mellin_transform 0.788515
logarithmic_derivative 0.785176
laplace_operator 0.784997
heaviside_step_function 0.783532
Top 30 analogous words or synonyms for weierstrass_transform
Article | Example |
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Weierstrass transform | The generalized Weierstrass transform provides a means to approximate a given integrable function arbitrarily well with analytic functions. |
Weierstrass transform | Setting "a"="bi" where "i" is the imaginary unit, and applying Euler's identity, one sees that the Weierstrass transform of the function cos("bx") is "e" cos("bx") and the Weierstrass transform of the function sin("bx") is "e" sin("bx"). |
Weierstrass transform | There is a formula relating the Weierstrass transform "W" and the two-sided Laplace transform "L". If we define |
Weierstrass transform | The above formal derivation glosses over details of convergence, and the formula "W" = "e" is thus not universally valid; there are several functions "f" which have a well-defined Weierstrass transform, but for which "e""f"("x") cannot be meaningfully defined. |
Weierstrass transform | Nevertheless, the rule is still quite useful and can, for example, be used to derive the Weierstrass transforms of polynomials, exponential and trigonometric functions mentioned above. |
Weierstrass transform | One may, alternatively, attempt to invert the Weierstrass transform in a slightly different way: given the analytic function |
Weierstrass transform | We can use convolution with the Gaussian kernel formula_13 (with some "t" > 0) instead of |
Weierstrass transform | In this context, rigorous inversion formulas can be proved, e.g., |
Weierstrass transform | where "x" is any fixed real number for which exists, the integral extends over the vertical line in the complex plane with real part "x", and the limit is to be taken in the sense of distributions. |
Weierstrass transform | The Weierstrass transform is intimately related to the heat equation (or, equivalently, the diffusion equation with constant diffusion coefficient). If the function describes the initial temperature at each point of an infinitely long rod that has constant thermal conductivity equal to 1, then the temperature distribution of the rod "t" = 1 time units later will be given by the function "F". By using values of "t" different from 1, we can define the generalized Weierstrass transform of . |