# Complex Numbers and Functions of a Complex Variable

• I. N. Bronshtein
• K. A. Semendyayev

## 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

## Notes

1. (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
2. (1).
For further details, see p. 593.Google Scholar
3. (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
4. (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
5. (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
6. (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
7. (1).
I.e., within an arbitrarily small circle with centre a except, possibly, the point a.Google Scholar
8. (2).
This case is analogous to that of a removable singularity of a function of a real variable (see p. 336).Google Scholar
9. (3).
In this case, we can find a sequence zn tending to a such that the sequence f(zn) will tend to an arbitrary preassigned complex number (except, at most, for one complex number).Google Scholar
10. (1).
The inversion with respect to a circle of the radius R is a mapping of the plane by which a point M1 lying at the distance d1 from the centre of the circle is sent into the point M2 lying on the same ray OM1 at the distance OM2 = d2 = R2/d1; thus the point M2 is sent into M1. The points lying inside the circle are mapped outside and conversely.Google Scholar
11. (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
12. (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
13. (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
14. (2).
See footnote on p. 601.Google Scholar
15. (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
16. (1).
17. (2).
The equation (1 + z2)3 = 0 has two 3-fold roots: i and − i, but only the first of them lies in the upper half-plane.Google Scholar