Introduction. Fundamental Concepts
There is hardly any body of theory which pervades modern mathematics as thoroughly as complex function theory. In itself remarkably harmonious, function theory lends order to such diverse fields as the theory of equations, conformal mapping and potential theory. It is also important for non-euclidean geometry and topology and for hydrodynamics, aerodynamics, electricity and thermodynamics. Altogether the theory of functions is still a live source of new mathematical discoveries.
Ever since the idea of function emerged as a basic concept of modern mathematical analysis mathematicians have been driven to extend the original notion by introducing complex variables. The new tool lent itself readily to formal computation, and mathematicians – though somewhat uncomfortable about the nature of these so-called "imaginaries" – did not disdain the power of their results. In the 18th century Leonard Euler, the unsurpassed master of analytical invention, observed that the power series representation of the exponential function \[e^{z}=1+\frac{z}{1 !}+\frac{z^{2}}{2 !}+\cdots\] yields the formula \[e^{i y}=\left(1-\frac{y^{2}}{2 !}+\frac{y^{4}}{4 !}-\cdots\right)+i\left(y-\frac{y^{3}}{3 !}+\frac{y^{5}}{5 !}-\cdots\right)\] or \[e^{iy}=\cos{y} + i \sin{y}\] by purely formal and uncritical substitution of the imaginary \(iy\) for \(z\) and regrouping of terms. Such methods led to other striking results, e.g. \[\arctan x=\frac{1}{2 i} \log \frac{1+i x}{1-i x}.\] This may be deduced from the series \[\log (1+z)=z-\frac{z^{2}}{2}+\frac{z^{3}}{3}-\cdots\] by substituting the values \(ix\) and \(-ix\) for \(z\) and then subtracting, the result being \(2i\) times the familiar series for \(\arctan{x}\). Not surprisingly, this uninhibited use of formal calculation occasionally led to paradox.1
Nonetheless, it was not until the 19\(^\text{th}\) century that this naive approach to mathematical analysis was replaced by the critical attitude of today. Functions of a complex variable were then studied systematically for the first time. The subsequent progress of mathematics has been largely in this field of function theory and the study of function theory has come to be regarded as the first step for any student of mathematics after he has mastered the elements of calculus.
This chapter covers: