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Argument (complex Analysis)
In mathematics (particularly in complex analysis), the argument of a complex number , denoted , is the angle between the positive real axis and the line joining the origin and , represented as a point in the complex plane, shown as \varphi in Figure 1. By convention the positive real axis is drawn pointing rightward, the positive imaginary axis is drawn pointing upward, and complex numbers with positive real part are considered to have an anticlockwise argument with positive sign. When any real-valued angle is considered, the argument is a multivalued function operating on the nonzero complex numbers. The principal value of this function is single-valued, typically chosen to be the unique value of the argument that lies within the interval . In this article the multi-valued function will be denoted and its principal value will be denoted , but in some sources the capitalization of these symbols is exchanged. In some older mathematical texts, the term "amplitude" wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number Illustration Modarg
Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each other * Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme such as power or status Complex may also refer to: Arts, entertainment and media * Complex (English band), formed in 1968, and their 1971 album ''Complex'' * Complex (band), a Japanese rock band * Complex (album), ''Complex'' (album), by Montaigne, 2019, and its title track * Complex (EP), ''Complex'' (EP), by Rifle Sport, 1985 * Complex (song), "Complex" (song), by Gary Numan, 1979 * "Complex", a song by Katie Gregson-MacLeod, 2022 * "Complex" a song by Be'O and Zico (rapper), Zico, 2022 * Complex Networks, publisher of the now-only-online magazine ''Complex'' Biology * Protein–ligand complex, a complex of a protei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Formula
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula states that, for any real number , one has e^ = \cos x + i \sin x, where is the base of the natural logarithm, is the imaginary unit, and and are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted ("cosine plus ''i'' sine"). The formula is still valid if is a complex number, and is also called ''Euler's formula'' in this more general case. Euler's formula is ubiquitous in mathematics, physics, chemistry, and engineering. The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics". When , Euler's formula may be rewritten as or , which is known as Euler's identity. History In 1714, the English mathematician Roger Cotes prese ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atan2
In computing and mathematics, the function (mathematics), function atan2 is the 2-Argument of a function, argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi 0, \\[5mu] \arctan\left(\frac y x\right) + \pi &\text x < 0 \text y \ge 0, \\[5mu] \arctan\left(\frac y x\right) - \pi &\text x < 0 \text y < 0, \\[5mu] +\frac &\text x = 0 \text y > 0, \\[5mu] -\frac &\text x = 0 \text y < 0, \\[5mu] \text &\text x = 0 \text y = 0. \end Instead of the tangent, it can be convenient to use the half-tangent as a representation of an angle, partly because the angle has a unique half-tangent, (See tangent half-angle formula.) The expression with in the denominator should be used when and to avoid possible loss of significance in computing . When an function is unavailable, it can be computed as twice the arctangent of the half-tangent . That is, |
Degree (angle)
A degree (in full, a degree of arc, arc degree, or arcdegree), usually denoted by ° (the degree symbol), is a measurement of a plane (mathematics), plane angle in which one Turn (geometry), full rotation is 360 degrees. It is not an SI unit—the SI unit of angular measure is the radian—but it is mentioned in the SI Brochure, SI brochure as an Non-SI units mentioned in the SI, accepted unit. Because a full rotation equals 2 radians, one degree is equivalent to radians. History The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year. Ancient astronomers noticed that the sun, which follows through the ecliptic path over the course of the year, seems to advance in its path by approximately one degree each day. Some ancient calendars, such as the Iranian calendar, Persian calendar and the Babylonian calendar, used 360 days for a year. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval (mathematics)
In mathematics, a real interval is the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real interval can contain neither endpoint, either endpoint, or both endpoints, excluding any endpoint which is infinite. For example, the set of real numbers consisting of , , and all numbers in between is an interval, denoted and called the unit interval; the set of all positive real numbers is an interval, denoted ; the set of all real numbers is an interval, denoted ; and any single real number is an interval, denoted . Intervals are ubiquitous in mathematical analysis. For example, they occur implicitly in the epsilon-delta definition of continuity; the intermediate value theorem asserts that the image of an interval by a continuous function is an interval; integrals of real functions are defined over an interval; etc. Interval ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Well-defined
In mathematics, a well-defined expression or unambiguous expression is an expression (mathematics), expression whose definition assigns it a unique interpretation or value. Otherwise, the expression is said to be ''not well defined'', ill defined or ''ambiguous''. A function is well defined if it gives the same result when the representation of the input is changed without changing the value of the input. For instance, if f takes real numbers as input, and if f(0.5) does not equal f(1/2) then f is not well defined (and thus not a function). The term ''well-defined'' can also be used to indicate that a logical expression is unambiguous or uncontradictory. A function that is not well defined is not the same as a function that is undefined (mathematics), undefined. For example, if f(x)=\frac, then even though f(0) is undefined, this does not mean that the function is ''not'' well defined; rather, 0 is not in the Domain of a function, domain of f. Example Let A_0,A_1 be sets, let A = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multi-valued
In mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a set-valued function with additional properties depending on context; some authors do not distinguish between set-valued functions and multifunctions, but English Wikipedia currently does, having a separate article for each. A ''multivalued function'' of sets ''f : X → Y'' is a subset : \Gamma_f\ \subseteq \ X\times Y. Write ''f(x)'' for the set of those ''y'' ∈ ''Y'' with (''x,y'') ∈ ''Γf''. If ''f'' is an ordinary function, it is a multivalued function by taking its graph : \Gamma_f\ =\ \. They are called single-valued functions to distinguish them. Motivation The term multivalued function originated in complex analysis, from analytic continuation. It often occurs that one knows the value of a complex analytic function f(z) in some neighbourhood of a point z=a. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Value Of Arg
Principal may refer to: Title or rank * Principal (academia), the chief executive of a university ** Principal (education), the head of a school * Principal (civil service) or principal officer, the senior management level in the UK Civil Service * Principal dancer, the top rank in ballet * Principal (music), the top rank in an orchestra Law * Principal (commercial law), the person who authorizes an agent ** Principal (architecture), licensed professional(s) with ownership of the firm * Principal (criminal law), the primary actor in a criminal offense * Principal (Catholic Church), an honorific used in the See of Lisbon Places * Principal, Cape Verde, a village * Principal, Ecuador, a parish Media * ''The Principal'' (TV series), a 2015 Australian drama series * ''The Principal'', a 1987 action film * Principal (music), the lead musician in a section of an orchestra * Principal photography, the first phase of movie production * "The Principal", a song on the album ''K-1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle \theta, the sine and cosine functions are denoted as \sin(\theta) and \cos(\theta). The definitions of sine and cosine have been extended to any real number, real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as Series (mathematics), infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic function, periodic pheno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sine
In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side opposite that angle to the length of the longest side of the triangle (the hypotenuse), and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle \theta, the sine and cosine functions are denoted as \sin(\theta) and \cos(\theta). The definitions of sine and cosine have been extended to any real value in terms of the lengths of certain line segments in a unit circle. More modern definitions express the sine and cosine as infinite series, or as the solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers. The sine and cosine functions are commonly used to model periodic phenomena such as sound and light waves, the posit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Periodic Function
A periodic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. The repeatable part of the function or waveform is called a ''cycle''. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called ''aperiodic''. Definition A function is said to be periodic if, for some nonzero constant , it is the case that :f(x+P) = f(x) for all values of in the domain. A nonzero constant for which this is the case is called a period of the function. If there exists a least positive constant with this property, it is called the fundamental period (also primitive period, basic period, or prime period.) Often, "the" period of a function is used to mean its fundamental period. A funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Turn (angle)
The turn (symbol tr or pla) is a unit of plane angle measurement that is the measure of a complete angle—the angle subtended by a complete circle at its center. One turn is equal to radians, 360 degrees or 400 gradians. As an angular unit, one turn also corresponds to one cycle (symbol cyc or c) or to one revolution (symbol rev or r). Common related units of frequency are '' cycles per second'' (cps) and '' revolutions per minute'' (rpm). The angular unit of the turn is useful in connection with, among other things, electromagnetic coils (e.g., transformers), rotating objects, and the winding number of curves. Divisions of a turn include the half-turn and quarter-turn, spanning a straight angle and a right angle, respectively; metric prefixes can also be used as in, e.g., centiturns (ctr), milliturns (mtr), etc. In the ISQ, an arbitrary "number of turns" (also known as "number of revolutions" or "number of cycles") is formalized as a dimensionless ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
