Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Computer algebra systemA computer algebra system (CAS) or symbolic algebra system (SAS) is any mathematical software with the ability to manipulate mathematical expressions in a way similar to the traditional manual computations of mathematicians and scientists. The development of the computer algebra systems in the second half of the 20th century is part of the discipline of "computer algebra" or "symbolic computation", which has spurred work in algorithms over mathematical objects such as polynomials.
EquationIn mathematics, an equation is a mathematical formula that expresses the equality of two expressions, by connecting them with the equals sign . The word equation and its cognates in other languages may have subtly different meanings; for example, in French an équation is defined as containing one or more variables, while in English, any well-formed formula consisting of two expressions related with an equals sign is an equation. Solving an equation containing variables consists of determining which values of the variables make the equality true.
Numerical methods for ordinary differential equationsNumerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly. For practical purposes, however – such as in engineering – a numeric approximation to the solution is often sufficient. The algorithms studied here can be used to compute such an approximation.
Hyperbolic angleIn geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of xy = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functions as coordinates. In mathematics, hyperbolic angle is an invariant measure as it is preserved under hyperbolic rotation. The hyperbola xy = 1 is rectangular with a semi-major axis of , analogous to the magnitude of a circular angle corresponding to the area of a circular sector in a circle with radius .
Transcendental functionIn mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function. In other words, a transcendental function "transcends" algebra in that it cannot be expressed algebraically. Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions. Formally, an analytic function f (z) of one real or complex variable z is transcendental if it is algebraically independent of that variable.
