Хэрэглэгч:Timur/Ноорог/Комплекс анализ

Комплекс анализ (буюу комплекс хувьсагчийн функцийн онол) нь комплекс тоон хувьсагчтай комплекс тоон утга авдаг функцүүдийг судалдаг математикийн салбар юм. Комплекс анализ нь тооны онол ба хэрэглээний математик; мөн түүнчлэн гидродинамик, термодинамик, ба цахилгааны инженерчлэл гэх мэт физикийн салбаруудад өргөн ашиглагддаг.

Комплекс анализийг математикийн хамгийн гоё бөгөөд хэрэглээтэй салбаруудын нэг гэдэгтэй ихэнх математикчид санал нийлдэг.

Комплекс анализад комплекс хувьсагчийн аналитик ба мероморф функцүүдийг түлхүү авч үздэг. Аналитик функцийн бодит ба хуурмаг хэсэг нь тус тусдаа Лапласийн тэгшитгэлийг хангадаг тул хоёр хэмжээстэй хавтгай дээрх физикийн бодлогуудыг бодоход комплекс анализ их хэрэглэгддэг.

Түүх

Комплекс анализ нь ерөнхийдөө 19-р зуунд суурь нь тавигдсан математикийн классик салбаруудын нэг юм. Энэ салбарыг хөл дээр нь босгосон хүмүүс гэвэл юуны түрүүнд Эйлер, Гаусс, Риман, Коши, Вейерштрасс нар дурдагдана. Комплекс анализийн уламжлалт хэрэглээ нь голдуу физик болон тооны аналитик онолд байдаг байсан бол сүүлийн үед голоморф буулгалтын давталтаар гарч ирдэг фракталуудаас болж комплекс анализийг сонирхох явдал их сэргэж байна. Комплекс анализийн өөр нэг чухал хэрэглээ бол конформ инвариант квант орны онол болох утасны онол юм.

Комплекс функц

A complex function is a function in which the independent variable and the dependent variable are both complex numbers. More precisely, a complex function is a function whose domain and range are subsets of the complex plane.

For any complex function, both the independent variable and the dependent variable may be separated into real and imaginary parts:

z=x+iy\,

and

w=f(z)=u(x,y)+iv(x,y)\,

where

x,y\in \mathbb {R} \,

and

u(x,y),v(x,y)\,

are real-valued functions.

In other words, the components of the function f(z),

u=u(x,y)\,

and

v=v(x,y),\,

can be interpreted as real-valued functions of the two real variables, x and y.

The basic concepts of complex analysis are often introduced by extending the elementary real functions (e.g., exponentials, logarithms, and trigonometric functions) into the complex domain.

Голоморф функц

Гол өгүүлэл: Holomorphic function

Holomorphic functions are complex functions defined on an open subset of the complex plane which are differentiable. Complex differentiability has much stronger consequences than usual (real) differentiability. For instance, holomorphic functions are infinitely differentiable, a fact that is far from true for real differentiable functions. Most elementary functions, including the exponential function, the trigonometric functions, and all polynomial functions, are holomorphic.

Гол үр дүнгүүд

One central tool in complex analysis is the line integral. The integral around a closed path of a function which is holomorphic everywhere inside the area bounded by the closed path is always zero; this is the Cauchy integral theorem. The values of a holomorphic function inside a disk can be computed by a certain path integral on the disk's boundary (Cauchy's integral formula). Path integrals in the complex plane are often used to determine complicated real integrals, and here the theory of residues among others is useful (see methods of contour integration). If a function has a pole or singularity at some point, that is, at that point its values "blow up" and have no finite boundary, then one can compute the function's residue at that pole, and these residues can be used to compute path integrals involving the function; this is the content of the powerful residue theorem. The remarkable behavior of holomorphic functions near essential singularities is described by Picard's Theorem. Functions which have only poles but no essential singularities are called meromorphic. Laurent series are similar to Taylor series but can be used to study the behavior of functions near singularities.

A bounded function which is holomorphic in the entire complex plane must be constant; this is Liouville's theorem. It can be used to provide a natural and short proof for the fundamental theorem of algebra which states that the field of complex numbers is algebraically closed.

An important property of holomorphic functions is that if a function is holomorphic throughout a simply connected domain then its values are fully determined by its values on any smaller subdomain. The function on the larger domain is said to be analytically continued from its values on the smaller domain. This allows the extension of the definition of functions such as the Riemann zeta function which are initially defined in terms of infinite sums that converge only on limited domains to almost the entire complex plane. Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface.

All this refers to complex analysis in one variable. There is also a very rich theory of complex analysis in more than one complex dimension where the analytic properties such as power series expansion still remain true whereas most of the geometric properties of holomorphic functions in one complex dimension (such as conformality) are no longer true. The Riemann mapping theorem about the conformal relationship of certain domains in the complex plane, which may be the most important result in the one-dimensional theory, fails dramatically in higher dimensions.

It is also applied in many subjects throughout engineering, particularly in power engineering.

Цааш үзэх

Ном хэвлэл

Gamelin, T.W., Complex analysis (Springer, 2001).
Shaw, W.T., Complex Analysis with Mathematica (Cambridge, 2006).
Needham T., Visual Complex Analysis (Oxford, 1997).
Henrici P., Applied and Computational Complex Analysis (Wiley). [Three volumes: 1974, 1977, 1986.]
Kreyszig, E., Advanced Engineering Mathematics, 9 ed., Ch.13-18 (Wiley, 2006).

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