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The Weierstrass's elliptic function is usually written with a rather special, lower case script letter ℘, which was Weierstrass's own notation introduced in his lectures of 1862–1863. [ footnote 1 ] It should not be confused with the normal mathematical script letters P: 𝒫 and 𝓅.
In mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function.
Weierstrass ℘-function. One of the most important elliptic functions is the Weierstrass ℘-function. For a given period lattice it ...
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve .
A classic example of a pathology is the Weierstrass function, a function that is continuous everywhere but differentiable nowhere. [1] The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions.
The definition for the gamma function due to Weierstrass is also valid for all complex numbers except non-positive integers: = = (+) /, where is the Euler–Mascheroni constant. [1] This is the Hadamard product of 1 / Γ ( z ) {\displaystyle 1/\Gamma (z)} in a rewritten form.
Among many other contributions, Weierstrass formalized the definition of the continuity of a function and complex analysis, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals.
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