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Function that is continuous everywhere but differentiable nowhere
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere
Weierstrass_function
Class of mathematical functions
mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class
Weierstrass_elliptic_function
Mathematical functions related to Weierstrass's elliptic function
mathematics, the Weierstrass functions are special functions of a complex variable that are auxiliary to the Weierstrass elliptic function. They are named
Weierstrass_functions
Mathematical theorem in the study of analysis
In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [a, b] can be uniformly
Stone–Weierstrass_theorem
Multifractal function used in terrain modeling and simulation
The Weierstrass–Mandelbrot function (often abbreviated W-M function and sometimes referred to as Weierstrass–Mandelbrot Noise) is a generalization of the
Weierstrass–Mandelbrot function
Weierstrass–Mandelbrot_function
German mathematician (1815–1897)
the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals. Weierstrass was born into
Karl_Weierstrass
Class of periodic mathematical functions
ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass ℘ {\displaystyle \wp } -function. Further development of this
Elliptic_function
Counterintuitive mathematical object
Weierstrass function, a function that is continuous everywhere but differentiable nowhere. The sum of a differentiable function and the Weierstrass function
Pathological_(mathematics)
Extension of the factorial function
an entire function, converging for every complex number z {\displaystyle z} . The definition for the gamma function due to Weierstrass is also valid
Gamma_function
"Smoothing" integral transform
mathematics, the Weierstrass transform of a function f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } , named after Karl Weierstrass, is a "smoothed"
Weierstrass_transform
Mathematical function whose derivative exists
function over the domain of f {\textstyle f} . Continuous functions may be nowhere differentiable in their domain, such as the Weierstrass function.
Differentiable_function
Mathematical functions
i{\bigr \}}.} The lemniscate functions and the hyperbolic lemniscate functions are related to the Weierstrass elliptic function ℘ ( z ; a , 0 ) {\displaystyle
Lemniscate_elliptic_functions
Class of functions behaving "like" periodic functions
the Weierstrass sigma function, which is quasiperiodic in two independent quasiperiods, the periods of the corresponding Weierstrass ℘ function. Bloch's
Quasiperiodic_function
Local theory of several complex variables
In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P. It
Weierstrass preparation theorem
Weierstrass_preparation_theorem
functions: The inverses of elliptic integrals; used to model double-periodic phenomena. Jacobi's elliptic functions Weierstrass's elliptic functions Lemniscate
List of mathematical functions
List_of_mathematical_functions
Change of variable for integrals involving trigonometric functions
substitutions introduced by Weierstrass to integrate rational functions of sine, cosine.) Two decades later, James Stewart mentioned Weierstrass when discussing the
Tangent half-angle substitution
Tangent_half-angle_substitution
Continuous function that is not absolutely continuous
derivatives at all rational numbers. Dyadic transformation Weierstrass function, a function that is continuous everywhere but differentiable nowhere. Vestrup
Cantor_function
Theorem in complex analysis
particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite)
Weierstrass factorization theorem
Weierstrass_factorization_theorem
Sochocki–Weierstrass theorem Stone–Weierstrass theorem Weierstrass–Enneper parameterization Weierstrass–Erdmann condition Weierstrass–Mandelbrot function Weierstrass
List of things named after Karl Weierstrass
List_of_things_named_after_Karl_Weierstrass
Topics referred to by the same term
continuous function on a closed and bounded set obtains its extreme values The Weierstrass–Casorati theorem describes the behavior of holomorphic functions near
Weierstrass_theorem
Fractal curve
drawing a tangent line to any point is impossible. Unlike the earlier Weierstrass function where the proof was purely analytical, the Koch snowflake was created
Koch_snowflake
Branch of mathematics studying functions of a complex variable
associated with complex numbers include Euler, Gauss, Riemann, Cauchy, Weierstrass, and many more in the 20th century. Complex analysis, in particular the
Complex_analysis
Instantaneous rate of change (mathematics)
nowhere. This example is now known as the Weierstrass function. In 1931, Stefan Banach proved that the set of functions that have a derivative at some point
Derivative
Function that is holomorphic on the whole complex plane
meromorphic function), then for entire functions there is a generalization of the factorization – the Weierstrass theorem on entire functions. Every entire
Entire_function
Special functions of several complex variables
quotients of the above four theta functions, and could have been used by him to construct Weierstrass's elliptic functions also, since ℘ ( z ; τ ) = − ( log
Theta_function
Algebraic curve in mathematics
numbers). The Weierstrass functions are doubly periodic; that is, they are periodic with respect to a lattice Λ; in essence, the Weierstrass functions are naturally
Elliptic_curve
Type of function in mathematics
an analytic function is a function that is locally represented by a convergent power series. More precisely, a real or complex function is analytic at
Analytic_function
Function which is not continuous at any point of its domain
irrational numbers and discontinuous at all rational numbers. Weierstrass function – a function continuous everywhere (inside its domain) and differentiable
Nowhere_continuous_function
Criterion about convergence of series
In mathematics, the Weierstrass M-test is a test for determining whether an infinite series of functions converges uniformly and absolutely. It applies
Weierstrass_M-test
Mathematical function with no sudden changes
published until the 1930s.[citation needed] Like Bolzano, Karl Weierstrass considered that a function y = f ( x ) {\displaystyle y=f(x)} at a point x = c {\displaystyle
Continuous_function
Mathematical function
and to define the Weierstrass transform. They are also abundantly used in quantum chemistry to form basis sets. Gaussian functions arise by composing
Gaussian_function
Functions of an angle
analysis, Wiley, pp. 315–316, LCCN 64-20061. Weierstrass, Karl (1841), "Darstellung einer analytischen Function einer complexen Veränderlichen, deren absoluter
Trigonometric_functions
Trigonometric functions: relate the angles of a triangle to the lengths of its sides. Nowhere differentiable function called also Weierstrass function: continuous
List_of_types_of_functions
Complex-differentiable (mathematical) function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood
Holomorphic_function
Functions in mathematics
the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R {\displaystyle f\colon U\to \mathbb
Harmonic_function
Point on a nonsingular algebraic curve
In mathematics, a Weierstrass point P {\displaystyle P} on a nonsingular algebraic curve C {\displaystyle C} defined over the complex numbers is a point
Weierstrass_point
Multivalued function in mathematics
In mathematics, the Lambert W function, also called the omega function or product logarithm, is a multivalued function, namely the branches of the converse
Lambert_W_function
Hyperbolic analogues of trigonometric functions
meromorphic in the whole complex plane. By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic
Hyperbolic_functions
Form of continuity for functions
continuous function f can fail to be absolutely continuous even on a compact interval. It may not be "differentiable almost everywhere" (like the Weierstrass function
Absolute_continuity
Wangerin functions Weber function Karl Weierstrass: Weierstrass function Louis Weisner: Weisner's method E. T. Whittaker: Whittaker function Wilson polynomial
List of eponyms of special functions
List_of_eponyms_of_special_functions
Theorem in transcendental number theory
Lindemann–Weierstrass theorem is a result that is very useful in establishing the transcendence of numbers. It states the following: Lindemann–Weierstrass theorem—if
Lindemann–Weierstrass_theorem
Mathematical curve whose shape is a fractal
the Mandelbrot set Menger sponge Peano curve Sierpiński triangle Weierstrass function The Beauty of Fractals Fractal antenna Fractal expressionism Fractal
Fractal_curve
Uniform restraint of the change in functions
shows uniformly continuous functions are not always differentiable. Despite being nowhere differentiable, the Weierstrass function is uniformly continuous
Uniform_continuity
Analytic function in mathematics
which may be used for a numerical evaluation of the zeta function. On the basis of Weierstrass's factorization theorem, Hadamard gave the infinite product
Riemann_zeta_function
Topics referred to by the same term
by sigma function one can mean one of the following: The sum-of-divisors function σa(n), an arithmetic function Weierstrass sigma function, related to
Sigma_function
Topics referred to by the same term
prime-counting function π(x), see Prime-counting function#Exact form. Almost nowhere differentiable Riemann function, on which the Weierstrass function has been
Riemann_function
Complex analysis theorem
simple curve in the plane, and φ {\displaystyle \varphi } an analytic function on C {\displaystyle C} . Note that the Cauchy-type integral ϕ ( z ) = 1
Sokhotski–Plemelj_theorem
Fractal curve resembling a blancmange pudding
blancmange curve. Cantor function (also known as the Devil's staircase) Minkowski's question mark function Weierstrass function Dyadic transformation Weisstein
Blancmange_curve
Fractal sets in complex dynamics of mathematics
(Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby
Julia_set
Replacing a number with a simpler value
the functions, however, is optional. Using the Gelfond–Schneider theorem and Lindemann–Weierstrass theorem, many of the standard elementary functions can
Rounding
Mode of convergence of a function sequence
first formalized by Karl Weierstrass. In 1821 Augustin-Louis Cauchy published a proof that a convergent sum of continuous functions is always continuous,
Uniform_convergence
Branch of mathematics
would not be until 150 years later when, due to the work of Cauchy and Weierstrass, a way was finally found to avoid mere "notions" of infinitely small
Calculus
1872 by Karl Weierstrass, and in fact examples had been found earlier of functions that were nowhere differentiable (see Weierstrass function). According
List_of_conjectures
Mathematical theorem
analysis, a branch of mathematics, the Casorati–Weierstrass theorem describes the behaviour of holomorphic functions near their essential singularities. It is
Casorati–Weierstrass_theorem
One-dimensional complex manifold
y)=(\wp (z),\wp '(z))} , where ℘ {\displaystyle \wp } is the Weierstrass elliptic function. Likewise, genus g {\displaystyle g} surfaces have Riemann surface
Riemann_surface
Weixiao (2018). "Hausdorff dimension of the graphs of the classical Weierstrass functions". Mathematische Zeitschrift. 289 (1–2): 223–266. arXiv:1505.03986
List of fractals by Hausdorff dimension
List_of_fractals_by_Hausdorff_dimension
Types of special mathematical functions
converges uniformly for all complex s and x. By a theorem of Weierstrass, the limiting function, sometimes denoted as γ ∗ {\displaystyle \gamma ^{*}} , γ
Incomplete_gamma_function
Major unsolved problem in transcendental number theory
for this more general result was given by Carl Weierstrass in 1885. This so-called Lindemann–Weierstrass theorem implies the transcendence of the numbers
Schanuel's_conjecture
Mathematical function
\sin } . The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions
Jacobi_elliptic_functions
Field of knowledge
accurate enough for avoiding paradoxes (non-Euclidean geometries and Weierstrass function) and contradictions (Russell's paradox). This was solved by the inclusion
Mathematics
Topics referred to by the same term
eta function may refer to: The Dirichlet eta function η(s), a Dirichlet series The Dedekind eta function η(τ), a modular form The Weierstrass eta function
Eta_function
Method of market analysis
Business cycle Demarcation problem The Wisdom of Crowds Kondratiev wave Weierstrass function Elliott, Ralph Nelson (1994). Prechter, Robert R. Jr. (ed.). R. N
Elliott_wave_principle
Point to which functions converge in analysis
Weierstrass's definition, a more general Heine definition applies to functions defined on subsets of the real line. Let f be a real-valued function with
Limit_of_a_function
Second-order partial differential equation
function. The Laplace operator therefore maps a scalar function to another scalar function. If the right-hand side is specified as a given function,
Laplace's_equation
Logarithm to the base of the mathematical constant e
to a multi-valued function: see complex logarithm for more. The natural logarithm function, if considered as a real-valued function of a positive real
Natural_logarithm
Mathematical function
modular forms. In particular the modular discriminant of the Weierstrass elliptic function with ω 2 = τ ω 1 {\displaystyle \omega _{2}=\tau \omega _{1}}
Dedekind_eta_function
Class of mathematical function
holomorphic function is constant, while there always exist non-constant meromorphic functions. Cousin problems Mittag-Leffler's theorem Weierstrass factorization
Meromorphic_function
Theorem
(w-a)^{n+1}}f(w)\right|\leq Mq^{n},} on C {\displaystyle C} , and as the Weierstrass M-test shows the series converges uniformly over C {\displaystyle C}
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
Index of lists with the same name
zeta function Other functions called zeta functions, but not analogous to the Riemann zeta function Jacobi zeta function Weierstrass zeta function Topics
List_of_zeta_functions
Countable intersection of open sets
0 , 1 ] ) {\displaystyle C([0,1])} . (See Weierstrass function § Density of nowhere-differentiable functions.) The notion of Gδ sets in metric (and topological)
Gδ_set
Mathematical function, denoted exp(x) or e^x
that the base e of the natural exponential function is a transcendental number, see the Lindemann–Weierstrass theorem. The Taylor series definition above
Exponential_function
Mathematical function
function, and a few software libraries provide it separately from the regular gamma function. Karl Weierstrass called the reciprocal gamma function the
Reciprocal_gamma_function
Type of continuity of a complex-valued function
It does not satisfy a Hölder condition of any order, however. The Weierstrass function defined by: f ( x ) = ∑ n = 0 ∞ a n cos ( b n π x ) , {\displaystyle
Hölder_condition
Mathematical functions which are smooth but not analytic
n\in \mathbb {N} } , this function is easily seen to be of class C∞, by a standard inductive application of the Weierstrass M-test to demonstrate uniform
Non-analytic_smooth_function
integer Elliptic function Abel elliptic functions Jacobi elliptic functions Lemniscate elliptic functions Weierstrass elliptic function Lee conformal world
Dixon_elliptic_functions
2.71828...; base of natural logarithms
Fourier's proof that e is irrational.) Furthermore, by the Lindemann–Weierstrass theorem, e is transcendental, meaning that it is not a solution of any
E_(mathematical_constant)
Formulas to determine the energy balance of a nonlinear wave
the Weierstrass ℘-function. This essentially follows because the three-wave interaction has exact solutions that are given by elliptic functions. The
Manley–Rowe_relations
About mathematical functions
a function as being defined by an analytic expression. In the 19th century, the demands of the rigorous development of analysis by Karl Weierstrass and
History of the function concept
History_of_the_function_concept
Mathematical series
metric for the space of functions that are added together in the series, and thus a different type of limit. The Weierstrass M-test is a useful result
Function_series
is not equal to the class of analytic function, then C M {\displaystyle C^{M}} does not satisfy the Weierstrass division property with respect to g (
Quasi-analytic_function
Extension of superfactorials to the complex numbers
written in terms of the double gamma function. Formally, the Barnes G-function is defined in the following Weierstrass product form: G ( 1 + z ) = ( 2 π
Barnes_G-function
non-Euclidean geometries in which the parallel postulate is wrong, the Weierstrass function that is continuous but nowhere differentiable, and the study by Georg
Philosophy_of_mathematics
Topics referred to by the same term
haplogroup W chromosome Lambert W function, a set of functions where w is any complex number Weierstrass function, a real function continuous everywhere but differentiable
W_(disambiguation)
Theory of a class of elliptic curves
ω 2 {\displaystyle \omega _{1},\omega _{2}} . Then we define the Weierstrass function of the variable z {\displaystyle z} in C {\displaystyle \mathbb {C}
Complex_multiplication
Method for the construction of fractals
S2CID 122674315. David, Claire (2019). "Fractal properties of Weierstrass-type functions". Proceedings of the International Geometry Center. 12 (2): 43–61
Iterated_function_system
Theorem in complex analysis
Liouville's theorem states that every bounded entire function must be constant. That is, every holomorphic function f {\displaystyle f} for which there exists a
Liouville's theorem (complex analysis)
Liouville's_theorem_(complex_analysis)
Strong form of uniform continuity
in it, an elementary consequence of the Stone–Weierstrass theorem (or as a consequence of Weierstrass approximation theorem, because every polynomial
Lipschitz_continuity
Concept of complex analysis
residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and
Residue_theorem
Fractal creation method
method of generating the attractor, or the fixed point, of any iterated function system (IFS). Starting with any point x0, successive iterations are formed
Chaos_game
Stochastic process generalizing Brownian motion
negative values on (0, ε). The function w is continuous everywhere, but nowhere differentiable (like the Weierstrass function). For any ϵ > 0 {\textstyle
Wiener_process
Analytic function that does not satisfy a polynomial equation
mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation whose coefficients are functions of the independent variable
Transcendental_function
sequence Function of a real variable Real multivariable function Continuous function Nowhere continuous function Weierstrass function Smooth function Analytic
List_of_real_analysis_topics
Compounding sum paid for the use of money
accumulation function is used instead. The accumulation function shows what $1 grows to after any length of time. The accumulation function for compound
Compound_interest
Power series with negative powers
Karl Weierstrass had previously described it in a paper written in 1841 but not published until 1894. The Laurent series for a complex function f ( z
Laurent_series
American mathematician
Oregon State University in 1955 for his dissertation on the inverse Weierstrass ℘ function. Besides his contributions in Fibonacci numbers and number theory
Verner_Emil_Hoggatt_Jr.
Blancmange curve Triflake[citation needed] Vicsek fractal von Koch curve Weierstrass function Z-order curve von Koch curve with random interval von Koch curve
List_of_mathematical_shapes
Concept in complex analysis
singularity of a complex-valued function of a complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity)
Zeros_and_poles
Function with two complex number "periods"
doubly periodic function with just one zero. Elliptic function Abel elliptic functions Jacobi elliptic functions Weierstrass elliptic functions Lemniscate
Doubly_periodic_function
Mathematical equation linking e, i and π
defined for complex z by extending one of the definitions of the exponential function from real exponents to complex exponents. For example, one common definition
Euler's_identity
Provides integral formulas for all derivatives of a holomorphic function
statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary
Cauchy's_integral_formula
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