Euler's formulaEuler'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 x: where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively. This complex exponential function is sometimes denoted cis x ("cosine plus i sine").
Exponential functionThe exponential function is a mathematical function denoted by or (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers.
Complex logarithmIn mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers. The term refers to one of the following, which are strongly related: A complex logarithm of a nonzero complex number , defined to be any complex number for which . Such a number is denoted by . If is given in polar form as , where and are real numbers with , then is one logarithm of , and all the complex logarithms of are exactly the numbers of the form for integers .
Taylor seriesIn mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.
Complex numberIn mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number , a is called the , and b is called the . The set of complex numbers is denoted by either of the symbols or C.
Slide ruleThe slide rule (also known colloquially in the United States as a slipstick) is a mechanical calculator (one of the simplest analog computers) hand-operated by sliding two rulers to perform multiplication and division primarily, and possibly exponents, roots, logarithms, and trigonometry. It is not typically designed for addition or subtraction, which is usually performed using other methods, like using an abacus. Maximum accuracy for standard linear slide rules is about three decimal significant digits, while scientific notation is used to keep track of the order of magnitude of results.
Complex planeIn mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers, and the y-axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors.
MultiplicationMultiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a product. The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier; both numbers can be referred to as factors.
AdditionAddition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or sum of those values combined. The example in the adjacent image shows two columns of three apples and two apples each, totaling at five apples. This observation is equivalent to the mathematical expression "3 + 2 = 5" (that is, "3 plus 2 is equal to 5").
Split-complex numberIn algebra, a split complex number (or hyperbolic number, also perplex number, double number) is based on a hyperbolic unit j satisfying A split-complex number has two real number components x and y, and is written The conjugate of z is Since the product of a number z with its conjugate is an isotropic quadratic form. The collection D of all split complex numbers for x,y \in \R forms an algebra over the field of real numbers. Two split-complex numbers w and z have a product wz that satisfies This composition of N over the algebra product makes (D, +, ×, *) a composition algebra.
Complex conjugateIn mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if and are real numbers then the complex conjugate of is The complex conjugate of is often denoted as or . In polar form, if and are real numbers then the conjugate of is This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: (or in polar coordinates).
Taylor's theoremIn calculus, Taylor's theorem gives an approximation of a -times differentiable function around a given point by a polynomial of degree , called the -th-order Taylor polynomial. For a smooth function, the Taylor polynomial is the truncation at the order of the Taylor series of the function. The first-order Taylor polynomial is the linear approximation of the function, and the second-order Taylor polynomial is often referred to as the quadratic approximation.
Complex analysisComplex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.
Brook TaylorBrook Taylor (18 August 1685 – 29 December 1731) was an English mathematician best known for creating Taylor's theorem and the Taylor series, which are important for their use in mathematical analysis. Brook Taylor was born in Edmonton (former Middlesex). Taylor was the son of John Taylor, MP of Patrixbourne, Kent and Olivia Tempest, the daughter of Sir Nicholas Tempest, Baronet of Durham. He entered St John's College, Cambridge, as a fellow-commoner in 1701, and took degrees in LL.B. in 1709 and LL.D. in 1714.
Matrix exponentialIn mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Lie group. Let X be an n×n real or complex matrix. The exponential of X, denoted by eX or exp(X), is the n×n matrix given by the power series where is defined to be the identity matrix with the same dimensions as .
Scalar multiplicationIn mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by a positive real number multiplies the magnitude of the vector—without changing its direction. The term "scalar" itself derives from this usage: a scalar is that which scales vectors.
Multiplication algorithmA multiplication algorithm is an algorithm (or method) to multiply two numbers. Depending on the size of the numbers, different algorithms are more efficient than others. Efficient multiplication algorithms have existed since the advent of the decimal system. If a positional numeral system is used, a natural way of multiplying numbers is taught in schools as long multiplication, sometimes called grade-school multiplication, sometimes called the Standard Algorithm: multiply the multiplicand by each digit of the multiplier and then add up all the properly shifted results.
Order of operationsIn mathematics and computer programming, the order of operations (or operator precedence) is a collection of rules that reflect conventions about which procedures to perform first in order to evaluate a given mathematical expression. For example, in mathematics and most computer languages, multiplication is granted a higher precedence than addition, and it has been this way since the introduction of modern algebraic notation. Thus, the expression 1 + 2 × 3 is interpreted to have the value 1 + (2 × 3) = 7, and not (1 + 2) × 3 = 9.
Polar decompositionIn mathematics, the polar decomposition of a square real or complex matrix is a factorization of the form , where is a unitary matrix and is a positive semi-definite Hermitian matrix ( is an orthogonal matrix and is a positive semi-definite symmetric matrix in the real case), both square and of the same size. Intuitively, if a real matrix is interpreted as a linear transformation of -dimensional space , the polar decomposition separates it into a rotation or reflection of , and a scaling of the space along a set of orthogonal axes.
Euler's identityIn mathematics, Euler's identity (also known as Euler's equation) is the equality where e is Euler's number, the base of natural logarithms, i is the imaginary unit, which by definition satisfies i2 = −1, and π is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is named after the Swiss mathematician Leonhard Euler. It is a special case of Euler's formula when evaluated for x = π. Euler's identity is considered to be an exemplar of mathematical beauty as it shows a profound connection between the most fundamental numbers in mathematics.
