Definitions

1. Functions are morphisms between sets, so this definition should be written using the forgetful functor that transforms an object into its underlying set.
2. Functions are a basic concept which appear in almost every academic discipline: In sociology, social functions are the basis of functionalism. (Web site)
3. Functions are defined using a define keyword and values are returned from them using a return followed by the return value in parentheses.
4. Functions are grouped by the datatypes of their arguments and their return values.
5. Functions are also objects, and support a large number of interesting mechanisms including composition, currying, mapping, and reduction. (Web site)

Various Functions

1. Functions of the liver The various functions of the liver are carried out by the liver cells or hepatocytes. (Web site)
2. The endocrine system chemically controls the various functions of cells, tissues, and organs through the secretion of hormones. (Web site)

Typical Examples

1. Typical examples of cryptographic primitives include pseudorandom functions, one-way functions, etc. (Web site)
2. Spaces of infinitely often differentiable functions defined on compact sets are typical examples. (Web site)
3. Typical examples of functions which are not holomorphic are complex conjugation and taking the real part.

Several Functions

1. In mathematics, several functions or groups of functions are important enough to deserve their own names.
2. Typical examples of analytic functions are: In mathematics, several functions are important enough to deserve their own name. (Web site)
3. Since the brain stem controls several functions, it is rarely the only part of the brain that is affected when it is injured.

Holomorphic

1. In science, many phenomena such as heat conduction and fluid flow are closely related to functions of complex numbers called holomorphic functions. (Web site)
2. Results obtained for sequences of holomorphic functions by Hartogs and Rosenthal in 1928 will be generalized for many other sheaves of functions. (Web site)
3. Complex analysis is particularly concerned with analytic functions of complex variables, known as holomorphic functions.

Transition Functions

1. Thus the underlying supermanifold possesses an atlas whose transition functions are $G^{infty}$ functions.
2. Definition 1 A manifold $M$ is called a complex analytic manifold (or sometimes just a complex manifold) if the transition functions are holomorphic.
3. It is also possible to define differentiability in terms of the transition functions. (Web site)

Cosine

1. If the sine and cosine functions are defined by their Taylor series, then the derivatives can be found by differentiating the power series term-by-term.
2. It is also sometimes useful to express the complex sine and cosine functions in terms of the real and imaginary parts of their arguments. (Web site)
3. Resolve vectors into components using the sine and cosine functions.

Sine

1. For functions where the derivative doesn't get more complex (like sine and cosine), you can make nmax as big as you like. (Web site)
2. All rotations are defined using the trigonometric "sine" and "cosine" functions. (Web site)
3. Polynomials serve to approximate other functions, such as sine, cosine, and exponential. (Web site)

Cosine Functions

1. Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions. (Web site)
2. Differentiating sin(x) and cos(x) from first principles In this unit we show how to differentiate the sine and cosine functions from first principles.
3. Fourier analysis - In morphometrics, the decomposition of an outline into a weighted sum of sine and cosine functions. (Web site)

Periodic Functions

1. Since the almost periodic functions do not form an algebra constructively, we may consider the f-algebra of functions generated by them. (Web site)
2. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves. (Web site)
3. His most noted contribution to mathematics was his formulation of the theory of almost periodic functions. (Web site)

Inverse Trigonometric Functions

1. Trigonometric functions and their graphs; trigonometric identities and equations; inverse trigonometric functions; solving triangles; complex numbers.
2. For inverse trigonometric functions, the notations sin − 1 and cos − 1 are often used for arcsin and arccos, etc. (Web site)
3. Analogous formulas for the other functions can be found at Inverse trigonometric functions. (Web site)

Basic Functions

1. Derivatives of basic functions (Calculus) Flash Tool [0] Take a quiz on derivatives (and a few antiderivatives) of exponential functions. (Web site)
2. Derivatives of basic functions (Calculus) Flash Tool [0] Take a quiz on derivatives (and a few antiderivatives) of polynomials. (Web site)
3. Derivatives of basic functions (Calculus) Java Applet Tool [1] Defines inverse trigonometric functions and explains their differentiations. (Web site)

Graphs

1. Arithmetic, algebra, geometry, trigonometry, functions and graphs, analysis.
2. Lambda calculus is a theory of functions as rules instead of graphs. (Web site)
3. Surfaces naturally arise as graphs of functions of a pair of variables, and sometimes appear in parametric form or as loci associated to space curves. (Web site)

Morphisms

1. When the objects are sets equipped with extra structure and properties, the morphisms are typically taken to be functions that preserve the extra structure. (Web site)
2. If morphisms in a category are viewed as analogous to functions, then morphisms in a multicategory are analogous to functions of several variables.
3. So to define a -functor we keep the function on objects and replace the functions between hom-sets with morphisms between hom-objects.

Complex Arguments

1. They can be extended to handle complex arguments in the completely natural way, so these functions are defined over the complex plane. (Web site)
2. These formulas can even serve as the definition of the trigonometric functions for complex arguments x. (Web site)
3. Inverse trigonometric functions can be generalized to complex arguments using the complex logarithm. (Web site)

Hyperbolic Functions

1. In mathematics, hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. (Web site)
2. The trigonometric and hyperbolic functions are also entire, but they are mere variations of the exponential function. (Web site)
3. Since the exponential function can be defined for any complex argument, we can extend the definitions of the hyperbolic functions also to complex arguments. (Web site)

Theta Functions

1. Many examples of such functions were familiar in nineteenth century mathematics: abelian functions, theta functions, and some hypergeometric series. (Web site)
2. So we see that the Theta functions can also be defined in terms of w and q, without a direct reference to the exponential function. (Web site)
3. Instead of expressing the Theta functions in terms of and, we may express them in terms of arguments and the nome q, where and. (Web site)

Several Complex Variables

1. On the removable singularities of functions of several complex variables.
2. Several complex variables is, naturally, the study of (differentiable) functions of more than one complex variable.
3. In mathematics, theta functions are special functions of several complex variables. (Web site)

Cryptographic Hash Functions

1. Cryptographic hash functions can be built using block ciphers. (Web site)
2. Being hash functions of a particular kind, cryptographic hash functions lend themselves well to this application too.
3. Note: To the date of writing of this document MD5 and SHA-1 are the most widely used cryptographic hash functions. (Web site)

Hash Functions

1. Functions with these properties are used as hash functions for a variety of purposes, both within and outside cryptography.
2. Another application is to build hash functions from block ciphers. (Web site)
3. Most widely used hash functions, including SHA-1 and MD5, take this form.

Structure

1. The Poisson bracket acts on functions on the symplectic manifold, thus giving the space of functions on the manifold the structure of a Lie algebra. (Web site)
2. The function of a protein is more directly a consequence of its structure rather than its sequence with structural homologs tending to share functions. (Web site)
3. A topological space is free of all imposed structure not relevant to the continuity of functions defined on it.

Sheaf

1. In the examples we gave above, the sections of the sheaf corresponded to functions. (Web site)
2. Rather, it emerges as a sheaf of functions as a consequence of the construction (via the quotients of local rings by their maximal ideals).
3. Sheaf of distributions Distributions on U The dual map to extension of smooth compactly supported functions by zero.

Stegun

1. Definitions of these functions and many of their properties can by found in Abramowitz and Stegun, Chapter 16-17.
2. Abramowitz and Stegun, handbook on special functions.
3. Hypergeometric functions are described in Abramowitz & Stegun, Chapters 13 and 15.

Mathematical Functions

1. For multiple-valued functions the branch cuts have been chosen to follow the conventions of Abramowitz and Stegun in the Handbook of Mathematical Functions. (Web site)
2. Handbook of mathematical functions with formulas, graphs, and mathematical tables, by Abramowitz and Stegun: National Bureau of Standards.
3. While based on C, NWScript does not have many functions from the C family, excluding logical and binary operators and some mathematical functions.

Inverses

1. Functions with left inverses are always injections. (Web site)
2. Logarithmic functions are the inverses of exponential functions. (Web site)
3. Elliptic functions: The inverses of elliptic integrals; used to model double-periodic phenomena. (Web site)

Elliptic Functions

1. Thus, functions that are periodic on a square grid, such as modular forms and elliptic functions, will thus inherit an SL(2, Z) symmetry from the grid. (Web site)
2. Elliptic functions are functions of two variables.
3. This is one area where we differ from Abramowitz and Stegun who use the modular angle for the elliptic functions.

Algebraic Functions

1. Examples of algebraic functions are rational functions and the square root function. (Web site)
2. The fields of algebraic functions of one variable appear in several areas of mathematics: complex analysis, algebraic geometry, and number theory. (Web site)
3. On a more significant theoretical level, using complex numbers allow one to use the powerful techniques of complex analysis to discuss algebraic functions. (Web site)

Rational Functions

1. One can manipulate polynomials in several variables over the integers, rational functions, and a variety of other mathematical objects. (Web site)
2. Algebraic functions that are neither polynomials nor rational functions may have one or more distinguishing characteristics.
3. In complex analysis, a partial fraction expansion is a way of writing a meromorphic function f(z) as an infinite sum of rational functions and polynomials.

Inverse Function

1. Sometimes, a function is described through its relationship to other functions (see, for example, inverse function).
2. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions. (Web site)
3. For certain functions, we can write a symbolic representation of the inverse function. (Web site)

Inverse Functions

1. Sometimes, a function is described through its relationship to other functions (for example, inverse functions). (Web site)
2. In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions. (Web site)
3. They learn the laws of sine and cosine, trigonometric functions and inverse functions, waves, conic sections, polynomial approximation, and much more. (Web site)

Implant

1. A dental implant looks and functions like a natural tooth, while having several advantages over other types of tooth replacement. (Web site)

Metal Screw

1. Calculus A dental implant is a metal screw that is placed into the jaw bone and functions as an anchor for a false tooth.
2. Calcium A dental implant is a metal screw that is placed into the jaw bone and functions as an anchor for a false tooth.

Functional Analysis

1. Such spaces, an essential tool of functional analysis and vector theory, allow analysis of classes of functions rather than individual functions. (Web site)
2. Functional analysis[ 7] studies spaces of functions and introduces concepts such as Banach spaces and Hilbert spaces.
3. In functional analysis, the functional derivative defines the derivative with respect to a function of a functional on a space of functions.

Differentiable Functions

1. However, powerful techniques from functional analysis have shown that these functions are in some sense "more common" than differentiable functions.
2. Differentiable Functions Having discussed continuity we will turn to another class of functions: differentiable functions.
3. Differentiation is a linear transformation from the space of all differentiable functions to the space of all functions. (Web site)

Integrals

1. Description: Vectors, lines, and planes, vector valued functions, partial derivatives, multiple integrals, calculus of vector fields. (Web site)
2. Differential equations Real analysis, the rigorous study of derivatives and integrals of functions of real variables.
3. He also wrote a book on calculus Calculus Calculus is a discipline in mathematics focused on limits, functions, derivatives, integrals, and infinite series.

Elementary Functions

1. Elementary functions of one complex variable: polynomials, exponential, logarithmic and trigonometric functions, their inverses. (Web site)
2. Elementary functions are functions built from basic operations (e.g.
3. Power series, elementary functions, Riemann surfaces, contour integration, Cauchy's theorem, Taylor and Laurent series, and residues.

Complex Variables

1. These developments are used to provide the basis for a theory of functions of real and complex variables.
2. Complex analysis is the study of functions of complex variables.
3. Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions).

Complex Variable

1. Runge's theorem has many applications in the theory of functions of a complex variable and in functional analysis.
2. We now present some major results in the theory of functions of a complex variable. (Web site)
3. This book discusses a variety of problems which are usually treated in a second course on the theory of functions of one complex variable.

Fourier Series

1. These topics include (amongst others): Infinite series and products, Sequences and series of functions and Fourier series.
2. The study of functions given by a Fourier series or analogous representations, such as periodic functions and functions on topological groups. (Web site)
3. Begins with a definition and explanation of the elements of Fourier series, and examines Legendre polynomials and Bessel functions. (Web site)

Special Functions

1. Topics include: differentiation, the Riemann-Stieltjes integral, sequences and series of functions, power series, Fourier series, and special functions. (Web site)
2. Topics include: differentiation, the Rieman integral, sequences and series of functions, uniform convergence, Taylor and Fourier series, special functions. (Web site)
3. One of the varieties of special functions which are encountered in the solution of physical problems is the class of functions called Legendre polynomials.

Exponential Function

1. Typical examples of entire functions are the polynomials, the exponential function, and sums, products and compositions of these. (Web site)
2. The exponential function e^x and the trigonometric function s sine and cosine are examples of such functions.
3. The polynomials and the exponential function e x and the trigonometric functions sine and cosine are examples of entire functions. (Web site)

Exponential Functions

1. Lines, algebraic functions, exponential functions, logarithmic functions, limits, derivatives, and integrals, with applications.
2. What types of functions don't have an inverse, like square root functions, exponential functions, cubic functions, etc. (Web site)
3. Logarithms: the inverses of exponential functions; useful to solve equations involving exponentials.

Complex Analytic Functions

1. Several complex variables also includes the study of the zero sets of complex analytic functions and these are called complex analytic varieties.
2. The fact that the class of complex analytic functions coincides with the class of holomorphic functions is a major theorem in complex analysis.
3. Study of such objects in one complex dimension is invariably boring as zero sets of complex analytic functions of one variable are just isolated points.

Smooth Functions

1. The smooth functions are those that lie in the class C n for all n; they are often referred to as C ∞ functions.
2. In mathematics, smooth functions (also called infinitely differentiable functions) and analytic functions are two very important types of functions.
3. In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions.

Polynomials

1. Quotients of polynomials are called rational expressions, and functions that evaluate rational expressions are called rational functions. (Web site)
2. Firstly, many functions which arise in pure and applied mathematics, such as polynomials, rational functions, exponential functions.
3. Polynomials are continuous functions as are functions represented by power series.

Trigonometric Functions

1. Familiar examples of functions with a single period on the real number line include the trigonometric functions like cosine and sine.
2. Trigonometric functions such as sine, cosine, and tangent are used in computations in trigonometry. (Web site)
3. All polynomials are continuous, and so are the exponential functions, logarithms, square root function and trigonometric functions. (Web site)

Several Variables

1. Functions of several variables: continuity, differentiability, partial derivatives, maxima and minima, Lagrange's method of multipliers, Jacobian. (Web site)
2. Differentiation of functions of several variables, change in the order of partial derivatives, implicit function theorem, maxima and minima. (Web site)
3. Functions of several variables including partial derivatives, multiple integrals, line and surface integrals. (Web site)

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