Continuous and piecewise continuously differentiable paths

L 0 IIJ

1. Continuous and piecewise continuously differentiable paths

According to 0.6.2 every continuous mapping 'y : I -+ C is called a path or a curve, with initial point -f (a) and terminal point y(b). Instead of y(t) the more suggestive notation z(t), or occasionally ((t), is also used. The path is called continuously differentiable or smooth if the function y is continuously differentiable on I.

Examples. 0) A path -y is called a null path if the function y is constant;

such paths are of course continuously differentiable.

1) The segment [zo, zl] from zo to zl is the continuously differentiable curve

z(t) :_ (1 - t)zo + tzl ^, t E [0,1].

2) Let c E C, r > 0. The function

z(t) := c+re`' = Rc+rcost+i($c+rsin t)

^, t E [a, b], where 0 < a < b < 21r, is continuously differentiable. The corresponding curve -y is called, as intuition dictates, a circular arc on the boundary of the disc Br(c). In case a = 0, b = 21r, -y is the circle of radius r around the center c. This curve is closed (meaning that initial point = terminal point); we designate this curve by Sr(c) or sometimes simply by S, and it is often convenient to identify S, with the boundary OB,.(c) of the disc Br(c).

If yr , ... , ym are paths in C and the terminal point of yµ coincides with the initial point of rye,+, for each 1 < p < m, then the path-sum y :_

y1 + y2 + + y,,, was defined in 0.6.2; its initial point is the initial point

of ry1 and its terminal point is the terminal point of ry,,,. A path -y is called piecewise continuously differentiable (or piecewise smooth) if it has the form -y = ryI +- +ry,,, with each 1M continuously differentiable. The figure shows such a path whose components rye, consist of segments and circular arcs.

Every polygon is piecewise continuously differentiable.

In the sequel we will be working exclusively with piecewise continuously differentiable paths and so we will agree that, from now on, the term "path"

will be understood to mean piecewise continuously differentiable path. Paths are then always piecewise continuously differentiable functions ry :[a, b]

C, that is, ry is continuous and there are points a1,. .. , am+l with a = a1 <

a2 < .. < am < am+1 = b such that the restrictions rye, := ryj [ai aja}1], 1 < !s < m are continuously differentiable.

2.

Integration along paths. As is 0.6.2,

¹ I = -y(I) designates the (compact) trace of the path ry. The trace of the circle Sr(c) is, e.g., the boundary of the disc BB(c) (cf. 0.6.5); we also write OBr(c) instead of Sr(c).

If ry is continuously differentiable, then f (z(t)) z'(t) E C(I) for every function f E C(f-(j); therefore according to 0.1 the complex number

r r

^b

J fdz :=

J

^{f (z)dz :=}

J

f (z(t))z'(t)dt

exists. It is called the path integral or the contour integral or the curvilinear integral of f E Qry1) along the continuously differentiable path ry. Instead of fy fdz we sometimes write f, fd( = fr f (C)d(. In the special case where -y is the real interval [a, b], described via z(t) := t, a < t < b, we obviously have

fdz =

jb

f (t)dt.

J

7

It follows that the integrals discussed in section 0 are themselves path integrals.

It is now easy to define the path integral f^yfdz for every path y =

ry1 + 'y2 + + rym for which the ryµ are continuously differentiable paths and for every function f E Qryl). We simply set

(*)

J fdz := >

µ=1 7P^'n

J

^fdz.

Note that each summand on the right side is well-defined because each ryµ is a continuously differentiable path with [-fµ[ C 1ryi.

Our contour integral concept is in an obvious sense independent of the par- ticular way -y is expressed as a sum; but to make this precise we would have to introduce some rather unwieldy terminology and talk about refinements of repre- sentations as sums. We will leave to the interested reader the task of formulating the appropriate notion of equivalence among piecewise continuously differentiable paths and proving that contour integrals depend only on the equivalence class of the paths involved.

3. The integrals f5B(t: - c)ndd. Fundamental to function theory is the

following

Theorem. For n E Z and all discs B = Br(c), r > 0,

JOB

([; - c)nd(_ ! 0 for n -1,

l 21ri

for n = -1.

Proof. Parameterize the boundary OB of B by C(t) := c + re't, with t E [0, 21r]. Then ('(t) = ire't and

JOB 0 0

Since n+1 e`(n+1)t isa primitive of ie'(n+l)t if n 0 -1, the claimed equality follows.

Much of function theory depends on the fact that faB(C - c)-1d( # 0.

The theorem shows that integrals along closed curves do not always van- ish. The calculation involved also shows (mutatis mutandis) that integrals along curves which each have the same initial point and each have the same terminal point need not be equal. Thus, e.g., (see the left-hand figure below)

= iri,

7+(-

2w 27f

( - c)ndC = J (re`t)nire'tdt =rn+1 J ie'(n+1)tdt.^.

- C

-

c = _1i.

In 1841 in his proof of LAURENT'S theorem WEIERSTRASS determined the value of the integral fSB C-1d( using a "rational parameterization" (cf. [W1), p.52):

He describes the boundary 8B of the disc (with c = 0) by means of C(t) := r ^, -oo < t < +oo. Evidently - as everyone used to learn in school - ((t) is the second point of intersection of 8B with the half-line starting at -r and having slope t := tan V (cf. the right-hand figure). Because

S' (t) = r

2i ('(t) - 2i

(1 - ti)2 ' C(t) 1 + t2 and it follows that

f

^{sB _ 2t 00}

^.1+t2

^{dt _ 4t dt1l+t2.}

WEIERSTRAss now defines (!) (which we proved above)

co 1

dt dt =

+t2 ⁴

, 1+t2'

J

^{o0 1}

-

the reduction to a proper integral being accomplished via the substitution t := 1

in f" 1

^. WEIERSTRASS remarked that all he really needed to know in his further deliberations was that this integral has a finite non-zero value. Cf. also 5.4.5 of the book Numbers [19].

4. On the history of integration in the complex plane. The first in-

tegrations through imaginary regions were published in 1813 by S. D. POISSON (French mathematician, professor at the Ecole Polytechnique). Nevertheless the first systematic investigations of integral calculus in the complex plane were made by CAUCHY in the two treatises [CiJ and [C2] already cited in the introduction to Chapter 1. The work [C1J was presented to the Paris Academy on August 22, 1814 but only submitted for printing in the "Memoires presentes par divers Sa- vants h l'Academie royale des Sciences de l'Institut de France" on September 14, 1825 and published in 1827. The second, essentially shorter work [C2] appeared as a special document (magistral memotre) in Paris in 1825. This document already contains the Cauchy integral theorem and is considered to be the first ex- position of classical function theory; it is customary and just (GAUSS' letter to BESSEL notwithstanding) to begin the history of function theory with CAUCHY'S treatise. Repeated reference will be made to it as we progress. CAUCHY was only gradually led to study integrals in the complex plane. His works make clear

that he thought a long time about this circle of questions: only after he'd solved his problems by separation of the functions into real and imaginary parts did he recognize that it is better not only not to make such a separation at all, but also to combine the two integrals

J(udx_vdv), J(vdx+ud),

which come up in mathematical physics in the study of two-dimensional flows of incompressible fluids, into a single integral

fdz with f:= u + iv , dz := dx + idy.

A good exposition of the development of the integral calculus in the complex plane along with detailed literature references is to be found in P. STACKEL's

"Integration durch Gebiet. Ein Beitrag zur Geschichte der Funktio- nentheorie," Biblio. Math. (3) 1(1900), 109-128 and the supplement to it by the same author: "Beitriige zur Geschichte der Funktionentheorie im achtzehnten Jahrhundert," Biblio. Math. (3) 2(1901), 111-121.

5. Independence of parameterization. Paths are mappings 'y : I -+ C.

You can think of ry as a "parameterization" of the trace or impression.

Then it is clear that this parameterization is somewhat accidental: one is inclined to regard as the same curves which are merely traversed in the same direction but in a different time interval or with different speeds. This can all be made precise rather easily:

Two continuously differentiable paths 7 : I - C, ry : I = [a, b] -+ C are called equivalent if there is a continuously differentiable bijection <p : I -+ I with everywhere strictly positive derivative gyp', such that y = ry o W.

The mapping cp is called a "parameter transformation" and is, because gyp' > 0, a strictly increasing function with a differentiable inverse. It then follows that W(a) = a, Wp(b) = b. The inequality cp' > 0 means intuitively that in parameter transformations the direction of progression along the curve does not change (no time-reversal!).

We immediately confirm that the equivalence concept thus introduced is a genuine equivalence relation in the totality of continuously differentiable paths. Equivalent paths have the same trace. We prove the important Independence theorem. If ry, ry are equivalent continuously differentiable paths, then

f fdz =

J

^fdz for every function f E C(I7I).

7 7

Proof. In the foregoing notation, i(t) = ry(W(t)) and sory'(t) = ry'(W(t))W'(t), t E I. It therefore follows that

fdz =

f

bf (7(t))7'(t)dt = f^bf (7(sv(t)))7

J

ry a a

According to Substitution Rule 0.2, applied to f (ry(t))ry'(t), the integral on the right coincides with f (a f (ry(t))i/(t)dt. Because sp(a) = a, Wp(b) = b, we consequently have ff fdz = fa f (ry(t))ry'(t)dt = fry fdz.

Thus the value of a path integral does not depend on the accidental parameterization of the path; and so, e.g., the Weierstrass parameterization (cf. 1.3) and the standard parameterization of BBr(0) both give the same values to integrals. Ideally we should from this point onward consider only equivalence classes of parameterized paths, even extending this idea in the natural way to piecewise continuously differentiable paths. But then every time we make a new definition (like sums of paths, the negative of a path, the length of a path) we would be obliged to show that it is independent of the class representative used in making it; the exposition would be considerably more unwieldy and complicated. For this reason we will work throughout with the mappings themselves and not with their equivalence classes.

6. Connection with real curvilinear integrals. As is well known, for a continuously differentiable path ry presented as z(t) = x(t) + iy(t), a < t < b, and real-valued continuous functions p, q on 1ryl, a real path integral is defined by

(Pdx + qdy)

J

6&(t), y(t))x'(t)dt + J q(x(t), y(t))y'(t)dt E R.

(_)

J

^{ry a a}

Theorem. Every function f

j(ud2

EC(II), with u := f, v :_ If, satisfies

J7

fdz = - vdy) + i

j(vdx

+ udy).

Proof. f = u + iv and z'(t) = x'(t) + iy'(t), so

f (z(t))='(t) = [u(x(t), y(t)) + iv(x(t), y(t))1 [x (t) + iy'(t)[

and the claim follows upon multiplying everything out and integrating.0 The formula in the statement of the theorem is gotten by writing dz = dx+idy, fdz = (u + iv)(dx + idy) and "formally" multiplying out the terms; cf also subsection 4.

One could just as easily have begun the complex integral calculus by using (*) to define the real integrals fV(pdx + qdy). It is entirely a matter of taste which avenue is preferred.

One can also introduce general complex path integrals of the form fY f dx, fY fdy, fY fdz for continuously differentiable paths ti and arbitrary f E C(1ryp, understanding by them the respective complex numbers

b b b

f f(z(t))x'(t)dt^,

f f(z(t))y

(t)dt , 1 f(z(t))z%(t)dt.

a a a

Then the identities

j ^fdx=(J

fdz+ J fdz) ^,

rfdy= 1

Ta

( J fdz-J fdz), rfdz= jfdz

Y Y

/

^{Y Y Y}

/

^{Y Y}

are immediate.

Exercises

Exercise 1. Consider the rectangle R := {z E C : -r < Rz < r , - s <

Sz < s}, whose boundary is the polygonal path

[-r-is,r-is]+[r-is,r+is]+[r+is,-r+is]+[-r+ is, -r -is),

where r > 0 and s > 0. Calculate f8R (_'CC.

Exercise 2. Let -y :

[0, 2ir] - C be y(t) := e't and let g :

171 -+ C be continuous. Show that

I g(()d( = -1

g(C)C2dC.

7

Exercise S. For a, b E R define ry : [0, 21r] - C by -y(t) := a cost + ib sin t and compute f, IKI2dt;.

§2 Properties of complex path integrals

The calculation rules from 0.1 carry over to path integrals; this will be our first order of business. With the help of the notion of the euclidean length of a curve we then derive (subsection 2) the standard estimate for path integrals, which is indispensable for applications. From it, for example, follow immediately (subsection 3) theorems dealing with interchange of limit and path integration.

1. Rules of Calculation. For all f,g E C(IryI), c E C

1) f,(f +g)dz = fry fdz+ f,gdz , f.1cfdz = cf. fdz.