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In mathematics, the complex plane or ''z''-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis. It can be thought of as a modified Cartesian plane, with the real part of a complex number represented by a displacement along the x-axis, and the imaginary part by a displacement along the y-axis.〔Although this is the most common mathematical meaning of the phrase "complex plane", it is not the only one possible. Alternatives include the split-complex plane and the dual numbers, as introduced by quotient rings.〕 The concept of the complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors. The multiplication of two complex numbers can be expressed most easily in polar coordinates—the magnitude or ''modulus'' of the product is the product of the two absolute values, or moduli, and the angle or ''argument'' of the product is the sum of the two angles, or arguments. In particular, multiplication by a complex number of modulus 1 acts as a rotation. The complex plane is sometimes called the Argand plane because it is used in Argand diagrams. These are named after Jean-Robert Argand (1768–1822), although they were first described by Danish land surveyor and mathematician Caspar Wessel (1745–1818).〔Wessel's memoir was presented to the Danish Academy in 1797; Argand's paper was published in 1806. (Whittaker & Watson, 1927, p. 9)〕 Argand diagrams are frequently used to plot the positions of the poles and zeroes of a function in the complex plane. ==Notational conventions== In complex analysis, the complex numbers are customarily represented by the symbol ''z'', which can be separated into its real (''x'') and imaginary (''y'') parts: : for example: ''z'' = 4 + 5''i'', where ''x'' and ''y'' are real numbers, and ''i'' is the imaginary unit. In this customary notation the complex number ''z'' corresponds to the point (''x'', ''y'') in the Cartesian plane. In the Cartesian plane the point (''x'', ''y'') can also be represented in polar coordinates as : In the Cartesian plane it may be assumed that the arctangent takes values from −''π/2'' to ''π/2'' (in radians), and some care must be taken to define the ''real'' arctangent function for points (''x'', ''y'') when ''x'' ≤ 0.〔A detailed definition of the complex argument in terms of the ''real'' arctangent can be found here.〕 In the complex plane these polar coordinates take the form : where :〔It can be shown (Whittaker & Watson, 1927, ''Appendix'') that all the familiar properties of the complex exponential function, the trigonometric functions, and the complex logarithm can be deduced directly from the power series for ''e''''z''. In particular, the principal value of log''r'', where |''r''| = 1, can be calculated without reference to any geometrical or trigonometric construction.〕 Here |''z''| is the ''absolute value'' or ''modulus'' of the complex number ''z''; ''θ'', the ''argument'' of ''z'', is usually taken on the interval 0 ≤ ''θ'' < 2''π''; and the last equality (to |''z''|''e''''iθ'') is taken from Euler's formula. Notice that the ''argument'' of ''z'' is multi-valued, because the complex exponential function is periodic, with period 2''πi''. Thus, if ''θ'' is one value of arg(''z''), the other values are given by arg(''z'') = ''θ'' + 2''nπ'', where ''n'' is any integer ≠ 0.〔(Whittaker & Watson, 1927, p. 10)〕 While seldom used explicitly, the geometric view of the complex numbers is implicitly based on its structure of a Euclidean vector space of dimension 2, where the inner product of complex numbers and is given by ; then for a complex number its absolute value || coincides with its Euclidean norm, and its argument with the angle turning from 1 to . The theory of contour integration comprises a major part of complex analysis. In this context the direction of travel around a closed curve is important – reversing the direction in which the curve is traversed multiplies the value of the integral by −1. By convention the ''positive'' direction is counterclockwise. For example, the unit circle is traversed in the positive direction when we start at the point ''z'' = 1, then travel up and to the left through the point ''z'' = ''i'', then down and to the left through −1, then down and to the right through −''i'', and finally up and to the right to ''z'' = 1, where we started. Almost all of complex analysis is concerned with complex functions – that is, with functions that map some subset of the complex plane into some other (possibly overlapping, or even identical) subset of the complex plane. Here it is customary to speak of the domain of ''f''(''z'') as lying in the ''z''-plane, while referring to the range or ''image'' of ''f''(''z'') as a set of points in the ''w''-plane. In symbols we write : and often think of the function ''f'' as a transformation of the ''z''-plane (with coordinates (''x'', ''y'')) into the ''w''-plane (with coordinates (''u'', ''v'')). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Complex plane」の詳細全文を読む スポンサード リンク
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