Portal:Analysis

Analysis

Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384–322 B.C.), though analysis as a formal concept is a relatively recent development.

The word comes from the Ancient Greek ἀνάλυσις (analysis, "a breaking up", from ana- "up, throughout" and lysis "a loosening").

As a formal concept, the method has variously been ascribed to Alhazen, René Descartes (Discourse on the Method), and Galileo Galilei. It has also been ascribed to Isaac Newton, in the form of a practical method of physical discovery (which he did not name).

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In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications. Trigonometric functions are commonly defined as ratios of two sides of a right triangle containing the angle, and can equivalently be defined as the lengths of various line segments from a unit circle. More modern definitions express them as infinite series or as solutions of certain differential equations, allowing their extension to arbitrary positive and negative values and even to complex numbers.

In modern usage, there are six basic trigonometric functions, which are tabulated here along with equations relating them to one another. Especially in the case of the last four, these relations are often taken as the definitions of those functions, but one can define them equally well geometrically or by other means and then derive these relations.

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In complex dynamics, the Julia set of a holomorphic function informally consists of those points whose long-time behavior under repeated iteration of ${\displaystyle f\,}$ can change drastically under arbitrarily small perturbations. Above is a 3D slice of a 4D Julia set.

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