A Guide Book to Mathematics pp 585-612 | Cite as

# Complex Numbers and Functions of a Complex Variable

Chapter

## Abstract

**Imaginary unit**. The *imaginary unit i*(^{1}) is formally defined as a number whose square is equal to − 1. Introducing the imaginary unit leads to a generalization of the concept of a number, namely, to the *complex numbers* which play an important role in algebra and analysis and also have real interpretations in certain geometric and physical problems.

## Keywords

Singular Point Complex Number Real Variable Imaginary Unit Coordinate Line
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## Notes

- (1).In electricity, the letter
*j*is used instead of*i*for the imaginary unit in order to avoid confusion with the notation*i*for current.Google Scholar - (1).For further details, see p. 593.Google Scholar
- (1).More generally, an
*algebraic function*can be defined implicitly by an equation \(a_{1}z^{m_1}\omega ^{n_1}+a_{2}z^{m_2}\omega ^{n_2}+\cdots +a_{k}z^{m_k}\omega ^{n_k}=0\), which may not be solvable explicitly (in terms of radicals) for ω.Google Scholar - (1).The convergence of a series at the points lying on the circumference of the circle of convergence requires additional examining in each particular case.Google Scholar
- (1).If a function ω =
*f*(*z*) is multiple-valued (as, for example, \(\sqrt[n]{z}\), Ln*z*, Arc sin*z*, Ar tanh*z*), then the domain of values of ω is a set of several or infinitely many surfaces placed one upon another, and to each value of the function there corresponds a point on one of the surfaces. These planes joined together along certain lines form the so-called*many-sheet Riemann surface*.Google Scholar - (1).In order that the condition shall be sufficient, we should require, in addition, that the partial derivatives involved in Cauchy-Riemann condition are continuous in the given domain.Google Scholar
- (1).
- (2).This case is analogous to that of a removable singularity of a function of a real variable (see p. 336).Google Scholar
- (3).In this case, we can find a sequence
*z*_{n}tending to a such that the sequence*f*(*z*_{n}) will tend to an arbitrary preassigned complex number (except, at most, for one complex number).Google Scholar - (1).The
*inversion*with respect to a circle of the radius*R*is a mapping of the plane by which a point*M*_{1}lying at the distance*d*_{1}from the centre of the circle is sent into the point*M*_{2}lying on the same ray*OM*_{1}at the distance*OM*_{2}=*d*_{2}=*R*^{2}/*d*_{1}; thus the point*M*_{2}is sent into*M*_{1}. The points lying inside the circle are mapped outside and conversely.Google Scholar - (1).The complex number in brackets following the name of a point is the value of the complex variable represented by this point.Google Scholar
- (1).For a simply connected domain see p. 342. In the case of a multi-connected domain, the condition may not be sufficient.Google Scholar
- (1).The radius of the inner circle can be equal to zero and then the annulus becomes a circle with the centre removed.Google Scholar
- (2).See footnote on p. 601.Google Scholar
- (3).In this case, the coefficients of the series are, by Cauchy’s formulas of p. 608, equal to \(c_{n}=\frac{1}{2\pi i}\int\limits_{\overleftarrow{C}}(\zeta -a)^{-n-1}f(\zeta )d\zeta =\frac{f^{(n)}(a)}{n!}\).Google Scholar
- (1).See p. 164.Google Scholar
- (2).The equation (1 +
*z*^{2})^{3}= 0 has two 3-fold roots:*i*and −*i*, but only the first of them lies in the upper half-plane.Google Scholar

## Copyright information

© Verlag Harri Deutsch, Zürich 1973