Holomorphy, angle and orientation conservation (final formulation) - 4. Cauchy, Riemann and Weierstrass views. The Cauchy Estimates for Derivatives in Disks - 2. The Gutzmer Formula and the Maximum Principle - 3. Historical Notes on the Cauchy Inequalities and the LIOUVILLE Theorem - 5*.
Izl>0and lzl=Oqz=0
A subset A of a metric space X is called dense in X if every nonempty open subset of X contains points A; this happens exactly when A = X. Recall that a set is called countable if the image N is under some map.].
Historical remarks on the convergence concept. Great difficul- ties attended the precise codification of this concept in the 19th century
A generalization of Riemann's rearrangement theorem was formulated in 1905 by P. Steinitz's substitution theorem is to be found, which is often used in linear algebra to prove .. the cardinality invariance of bases and which is a very scary exam question. on p. 133 of the first part of this work.). The following statements about a Banach space V are equivalent: .. i) The class of unconditionally convergent series E v, v E V, coincides with the class of absolutely or normally convergent ones, i.e., those for.
Historical remarks on the concept of continuity. LEIBNIZ and
An interesting research that brings this up to date is Dieter RuTHING's "Some definitions of the concept of function from Joh.
Examples involving the Cauchy-Riemann equations
In the last two examples, complex-differentiable functions were made out of transcendental real functions using the Cauchy-Riemann equations. For every twice real differentiable function u : D -' R, the function Du = uxx + uyy is defined in D. u is called a potential function in D if u satisfies the potential equation Du = 0 in D. language is motivated by considerations from physics, especially electrostatics because functions with Du = 0 arise as potentials in physics.). The essence of the theorem is that the real and imaginary parts of complex differentiable functions are potential functions.
For more details on the differential calculus with respect to z and 2, see subsection 4. Cauchy-Riemann differential equation 8f/82=0. The latter can now also be written as the condition for holomorphy can be compressed into the single equation ifx = fy.
C,t-C+y'(t;)t, tER
- hA=ide*A=aEforsome aEC"
- Ihc(z)12 = iI2
- The exponential and trigonometric series. Euler's formula
Such pathologies are eliminated by introducing the idea of locally uniform convergence. The theory of uniform convergence becomes especially transparent with the introduction of the highest semi-norm. The history of the concept of uniform convergence is a paradigm in the history of ideas in modern mathematics.
The realization of the central role of the concept of uniform convergence in analysis emerged slowly over the past century. The quantity RE [0, oo], determined by the convergence theorem, is called the radius of convergence, and the set BR is called the convergence disk (sometimes less accurately, the convergence circle) of the power series. Using the ratio criterion we will determine the radii of convergence of some important power series.
We speak of the complex exponential function and the complex cosine and sine functions.
L 0 IIJ
Historical remarks on term-wise differentiation of series. For
EULER It was self-evident that the differentiation of power series and functional series by terms led to the derivation of the limit function. The correct function-theoretic generalization of the theorem on the factorial differentiation of power series is the famous WEIERSTRASSE theorem on the factorial differentiation of compactly convergent series of holomorphic functions. This important differential equation can be the starting point for the exponential function theory (cf. 3) Cosine function and sine function.
The lemma and the theorem of units both of course also apply to formal power series; in this case, the proof of the lemma simply reduces to the first two lines of the above proof. An element p of an integral domain R is called a prime element if it is not a unit of R and if every time f, g E R and p divide the product f g, then p divides one of the factors f or g.
Determination of all ideals. A commutative ring R is called a
The ring A is factorial and until multiplication by units the element z is the only prime number in A. In the center is the exponential function, which is determined (§1) both by its differential equation and by its addition theorem. Following Euler's recommendation, all important properties of the trigonometric functions are derived from the exponential function via the.
In particular, we will see that ir is the smallest positive zero of the sine function and z is the smallest positive zero of the cosine function, just as we learned in infinitesimal calculus. This last property, together with exp 0 = 1, characterizes an exponential function; allows a very elegant derivation of the basic properties of this function.
Characterization of exp x by its differential equation. First let us note that
Logarithm functions will be discussed in detail in §4 and §5, where the general power function and the Riemann zeta function will also be introduced. The most important holomorphic function that is not a rational function is defined by the power series E 7 and denoted by exp z. Its dominant role in the complex theater is due to the Euler formulas and to its invariance under differentiation: exp' = exp.
Decisive in many arguments is the elementary fact that the holomorphic function f with f' = 0 is necessarily constant.
The addition theorem of the exponential function. For all z and w in C
Since this is the case for any z E G, the function e is holomorphic in G and the current theorem is a consequence of the previous one. With this definition of the symbol "cubit" the addition theorem reads like a power rule or law of exponents. The use of the usual formula for the sum of a finite geometric series and another application of the addition theorem led to.
These kinds of conclusions aroused quite a bit of admiration in Euler's time; HADAMARD is said to have said of this phenomenon: "Le plus court chemin entre deux enoncr s reels passe par le compleze (The shortest path between two statements about reals is through complexes)."
As in the case of the corresponding addition theorems for real arguments, countless further identities flow from these two. These functions were discovered by geometers long before the advent of the exponential function; a formula closely related to the addition theorem for sin(a+Q) and sin(a-/3) was already known to ARCHIMEDES. The mapping exp : C --+ C' is a group homomorphism of the additive group C of all complex numbers into the multiplicative group C" of all non-zero complex numbers.
As far as this book is concerned, the content of the above statement is the definition of rr. Based on this discussion, a simple visualization of the exponential function is possible: divide the z-plane into infinitely many continuous strips.
Course of values, zeros, and periodicity of cos z and sin z
Real numbers Exl, V are called polar coordinates z; the number W is called the argument z. The unit disc E is the region of the holomorphy of the function h(z) := F, 2-"z2° and this function (unlike the previous g) is continuous on E=EUBE. In other words -L(i-P) is a continuous integer-valued function on G. .. the image of the connected set G is therefore a connected subset of Z, i.e. one point, say n.
0, e1<' 1= 1 holds and therefore, according to the derived characterization of the previous subsection, log is a logarithmic function of B1(1). The logarithm function in the slot plane C- just introduced is called the main branch of the logarithm; for that log i = 17ri.
Historical remarks on logarithm functions in the complex do- main. The extension of the real logarithm function to complex arguments
In contrast, Z is not a logarithmic function in the left half-plane at all; since obviously exp(t(z)) = -z. One could easily remove any other half-line starting at 0 and define a logarithmic function in the resulting region by procedures analogous to those above. But in the whole of C" there is no logarithmic function; for any such function would have to coincide in C- with some branch of log z + 21rin, and consequently would not be continuous at every point of the negative real axis.
Based on the permanence principle, according to which all functional relations that hold in the real domain must continue to hold in the complex domain, people as late as the beginning of the 18th century believed in the existence of a (unique) function log z which satisfies the equations. In the complex domain, this equation no longer enjoys unlimited validity, since exp : C --+ C" is not injective.
On the identities log(wz) = log to + log z and log(exp z) = z
The real logarithm function is often introduced as the inverse (real) exponential function, and so log(exp x) = x for x E R. The strip Go = {z E C : -7r < Qrz < 7r} is mapped biholomorphically (and thus certainly topologically ) to the alit-plane C- with an exponential function, and the inverse mapping is the principal branch of the logarithm. This formula can be used to explicitly calculate the value of a binomial series.
But the progression of x-values can take place in an infinite number of ways: just as we think of the realm of all real quantities as an infinite straight line, so we can imagine the realm of all quantities, real and imaginary, as an infinite plane in which each point determined by an abscissa a and an ordinate b also represents the quantity a + bi. The primary tool in these studies is the Fundamental Theorem of Differential and Integral calculus on real intervals (cf. 0.2).
The integral concept. Rules of calculation and the standard
This usage is suggested by thinking of the definition of the integral in terms of Riemann sums. From that point of view, the inequality just established indeed generalizes the A-inequality 1w + zI < IwI + IzI for complex numbers.
The fundamental theorem of the differential and integral cal- culus. For calculating integrals the Fundamental Theorem of Calculus is
The proof consists of going into real and imaginary parts, applying the fundamental theorem of calculus for real-valued functions, and reassembling the parts. In all important applications on the other hand, we only need to integrate over paths consisting of line segments and circular arcs connected together. If, however, we consider the wider class of all different piecewise continuous paths, the reason is not very frequent and pedagogic.
There was a time when it was fashionable to sacrifice valuable class time developing the most general theory of line integrals. Nowadays it is more common in lectures on basic function theory to limit themselves to integration along piecewise continuously differentiable curves and move on to the main points of the theory.
Continuous and piecewise continuously differentiable paths
In what follows, we will work exclusively with piecewise continuous differentiable paths, and so will, from now on, agree to the term "path". It is called the path integral or the contour integral or the curve integral of f E Qry1) along the continuously differentiable path ry. We will leave to the interested reader the task of formulating the appropriate notion of equivalence between piecewise continuously differentiable paths and proving that the contour integrals depend only on the equivalence class of the paths involved.
We immediately confirm that the concept of equivalence introduced in this way is a true equivalent relation on the set of continuously differentiable paths. Ideally, from this point on we should consider only the equivalent classes of parameterized paths, even extending this idea naturally to piecewise continuously differentiable paths.
