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In the electromagnetic theory of harmonic waves in linear systems, the various field quantities are often represented by complex numbers which contain the time t through a factor eiωt. Then the electric and magnetic field vectors are complex vectors, and in general may be thought of as having complex lengths and directions. If the instantaneous field associated with a complex vector E is drawn as a real vector with a fixed origin, its end point traverses an ellipse whose features can be expressed in terms of the scalar and vector products of E with itself or with its complex conjugate. Some applications of this to the design of aerials for space vehicles, and to the relations between the electric and magnetic fields in magnetoionic theory are discussed.

(1917). The Graph of F(X) for Complex Numbers. The American Mathematical Monthly: Vol. 24, No. 9, pp. 409-420.

The paper contains 2 sections. In section 1 the spectral theory for rectangular complex matrices, as given by Penrose and Hestenes is reformulated so as to include the spectral theory for normal matrices as a special case. These results, which are mostly well known, are used in section 2 to study matric functions of rectangular complex matrices. These functions are shown in theorem 4 to satisfy an appropriate version of the Fantappie requirements. Theorems 5 and 6 are the analogs of Cauchy's integral theorem and Taylor's theorem respectively. (Author)

Multiparameter systems with dominant complex roots constrained by degrees of freedom to construct reduced characteristic equation

This paper presents a clear interpretation of the pole migration on the real axis where many root loci starting from different poles lie together or conflict with one another, by introducing the concept of a pair of real roots. Except in the case of a single root locus, the root loci on the real axis appear always as a pair. Just like the complex number, a pair of real roots is expressed in two parts and the whole root loci for both real and complex roots are plotted in the conventional rectangular coordinates as continuous curves, by which the migration of each pole on the real axis is shown very clearly. Three examples are given.

(1929). Some Geometrical Applications of Complex Numbers. The American Mathematical Monthly: Vol. 36, No. 10, pp. 504-511.

A complex number is defined which characterizes the behaviour of the velocity vector in an n-dimensionnl state space. Methods are then developed for the mapping of state space velocity fields into the complex plane. These techniques should be useful in the design of multivariable feedback control systems, and a simple example of their application is given.

Bellman et al. (1956) found the optimal control function for an nth order control system characterized by a stability matrix having distinct real characteristic roots. In this paper the theory has been extended to include stability matrix having either repeated real roots or non-repeated complex roots. Two examples have been treated to illustrate the use of the theory.

Summary A method is described for the evaluation of the complex roots of algebraic equations in which the roots are evaluated directly from the original equation by a process somewhat akin to Homer's method for real roots.

(1966). Automorphisms of the Complex Numbers. Mathematics Magazine: Vol. 39, No. 3, pp. 135-141.

(1967). On the Automorphisms of the Complex Number Field. Mathematics Magazine: Vol. 40, No. 4, pp. 213-213.

The purpose of this article is to present an elementary introduction to complex numbers with a glance at some applications. The intent is to present the concepts intuitively, more or less as they probably originated, rather than formally by -a rigorous procedure of definitions and postulates. Inevitably this leads to some repetitions in statements but in an elementary presentttion this is not a pedagogical fault. A brief review of the history of complex numbers is interesting and relevant. The first introduction of imaginary numbers occurred in connection with the solution of cubic equations. During the 16th century Italian mathematicians of the then famous university of Bologna discovered that sometimes correct answers could be obtained moreexpeditiously if they assumed 1) a symbol, i, 2) i2 = -1, and 3) in other respects treated i as an ordinary number. One of QCardano's problems was "Divide 10 into two parts whose product is 40." Cardano was intrigued by the answer but at the same time he was very doubtful of the method. For several hundred years mathematicians continued to play -with the concept. Euler (1707-1783) achieved brilliant results by the use of complex numbers but the fundamental principles of their logic was either not deemed important or was completely misunderstood. Struik ways that in the 18th century 'There was experimentation with infinite series, with infinite products, ... With the use of symbols such as 0, OD, 0,j?, ... Much of the work of the leading mathematicians impresses us as wildly enthusiastic experimentation." Wessel, a Danish mathematician, probably furnished the first logical foundation for complex numbers, but his paper 'On the Analytical Representation of Direction," presented in 1797, was unknown to mathematicians until republished one hundred years later. Argand, a Paris accountant, also presented in 1806 a logical foundation but his work appears to have received very little attention. Actually Gauss (17771855) first used the phrase complex number, the symbol i forthe imaginary unit, and introduced mathematicians to the true theory of these numbers. Gauss' work was followed by that of Cauchy (1789-1857) and that of Riemann (1826-1866). From their works has arisen the complex function theory which is basic and indispensable for advanced mathematics, and even for a full comprehension of the more elementary theorems of algebraic analysis. The point to be emphasized is this. More than 200 years passed before mathematicians placed complex numbers upon a firm logical basis.

The only place in a course in real differential equations that complex numbers arise (embarrassingly, I'm afraid) is in the study of the linear equation with constant coefficients. This seems to imply that the theory is not complete in itself, without trespassing into -the complex number field. The purpose of this paper is to show that one need -not introduce complex quantities at all to obtain the solution even when the auxiliary equation does not have real roots. The equation we-are concerned with is

vative (-.,)It T. of the tensor T::: as a gemetrical genera6t t0 lization of EL. Post' s deri vati ve { f (D)}t such that if f (D) DM to0 where M is an arbitrary complex number and particularly a negative integer -m it gives the M-th intrinsic derivative as a generalization of the fractional derivative and the m-th order tensorial integral of tensors respectively. For this purpose, at first, we shall show that the excovariant and crossed extensors of integral infinite ranges are derived from tensors by Post' s differentiation and then, from such generalized derived extensors, we construct the required derivatives of covariant tensors in first a nonholonomic space whose original space is affinely connected and treat of their properties.