On the identities log(wz) = log to + log z and log(exp z) = z

L 0 IIJ

1. On the identities log(wz) = log to + log z and log(exp z) = z

For numbers w,z and wz in C-, with w = Iwle'', z = lzle'l', wz =

lwzle'X, where co,?/i, X E (-ir, a), there is an n E {-27r, 0, 2ir} such that X = cp + ij' + rl. From this it follows that

log(wz) = log(lwIlzl) + iX = (log 1wJ + iw) + (log IzI + iv,) + ill

= loges+logz+irl.

We see in particular that

log(wz) = log w + log z t* cp + 10 E (-7r, 9r).

Since the condition -ir < cp + i/i < it is met whenever R w > 0 and Rz > 0, a special case of the above is

log(wz) = log w+log z for all w, z E C with R w > 0, $tz > 0.

0

The number log(exp z) lacks definition precisely for those z = x + iy for which e= = ex cos y + iex sin y falls into C \ C-. This happens exactly when ex cosy < 0 and e= sin y = 0, that is, when y = (2n + 1)7r for some n E Z.

Therefore logo exp is well-defined in the domain

B:=C\{z:£z=(2n+1)ir,nEZ}=UGn,

nEZ

where for each n E Z

Gn := {z E C : (2n - 1)ir < £ z < (2n + 1)7r},

by

13ai

xi

-si

a strip of width 2ir parallel to the x-axis (cf. the figure above).

For z = x + iy E Gn we have ex = e=e$(V_2nr) andy - 2nir E (-ir,1r).

It follows that

log(exp z) = log eZ + i(y - 2n7r) and so

log(exp z) = z - 27rin for all z E Gn, n E Z.

Only in the strip Go does log(exp z) = z prevail. Since however exp(log z) _ z always holds, we have

The strip Go = {z E C : -7r < Qrz < 7r} is mapped biholomorphically (and so certainly topologically) onto the alit-plane C- by the exponential function, and the inverse mapping is the principal branch of the logarithm.

2. Logarithm and arctangent. The arctangent function, defined in the unit disc in 2.5 satisfies

1 + iz z

(1) arctan z =1. log

2i

1-iz

^{E E.}

Proof. The function h(z)

' E O(C \ {1}) is, to within the factor i,

the Cayley mapping hC' from 2.2.2; therefore h(E) = {z E C : Rz > 01.

Accordingly, the function H(z) := log h(z) is well-defined in E and lies

in O(E). It satisfies H(z) =

^{h 4.} Therefore G(z) := H(iz) - 2 arctan z E O(E) satisfies

G'(z) = iH'(iz) - 2i(arctan)'(z) =

1 +1z2

- 1

+tzs = 0

for all z E E. Since G(0) = 0, it follows that G - 0.

0

On the basis of the identity arctan(tan z) = z proved in 2.5, equation (1) yields

1

1+itanz

(2)

z = - log

for all z near 0.

2i 1 - i tan z We further infer that

(3) tan(arctan z) = z for z E E.

Proof. w := arctan z satisfies e21r = ^{1 + ix}

1iz

, on account of (1). Since

1-e2w

tan w = i_{1 +}e21w (cf. 2.5), it follows that tan w = z.

3. Power functions. The NEWTON-ABEL formula. As soon as a logarithm function is available, general power functions can be introduced.

If 1: G - C is a logarithm function, we consider the function p, : G -+ C , z '- exp(ae(z)).

for each complex number a. We call p, the power function with exponent a based on e. This terminology is motivated by the following easily verified assertions:

Every function p, is holomorphic in G and satisfies p', = op,_ 1. For all a, T E C, pp, = po+r and for n E N, p (z) = z' throughout G.

In the slit-plane C- a power function with exponent a is defined by exp(a log z). Except for a brief interlude in 14.2.2, we reserve the (some- times dangerously seductive) symbol z° for this power function. For whole numbers a E Z this agrees with the usual notation and, as remarked above, is consistent with the prior meaning of that notation. We have, for example,

1° = 1 , it = e-f 0.2078795763...

Remark. That i' is real was remarked by EULER at the end of a letter to GOLDBACH of June 14, 1746: "Letztens habe gefunden, dab these expressio einen valorem realem habe, welcher in fractionibus decimalibus

= 0,2078795763, welches mir merkwiirdig zu seyn scheinet. (Recently I have found that this expression has a real value, which in decimal fraction form = 0.2078795763; this seems remarkable to me.)" Cf. p. 383 of the "Correspondence entre Leonard Euler et Chr. Goldbach" cited in

1.3.

The rules already noted above can be suggestively written in the new notation thus:

(zo)' = ^{az0-1 ,} zozr = Za+r ,

z E C_-

From the defining equation z° = e° log `, z E C-, follows:

For z = re' , V E (-a, ir), and o = s + it, we have Jz°I and so Iz°I < JzI*0e"I10I.

Proof. All is clear because le'l = emw and Ie-0001 < e"lu°I.

The function (1 + z)° is, in particular, well defined in the unit disc E.

Since (1 + z)° = exp(alog(1 + z)) = b°(z) according to 4.3.4(*), we have the following

NEWTON-ABEL formula:

00 / \

(1+z)°=E1 ° Iz° forallaEC,zEE.

By means of this formula the value of the binomial series can be explicitly calculated. Setting a = s + it and 1 + z = re'V, we have

b°(z) = exp(°log(l + z)) = e(,+it)(log r+iio) = r,e-t'Pe:(tlogr+,w) If you write z = x + iy, comparison of real and imaginary parts on both sides of the equation 1 + x + iy = re''° yields r = ((1 + x)2 + y2)1/2,

= arctan 1, and consequently

(1 + z)° = ((l +x)2 + y2)Jae-tarctan T x x

[cos(sarctanj_L-

1

+ x+ t log((1 + x)2 +y2)) + i sin I a arctan

l +

_x+ 2t log((1 + x)2 +^y2)

)

J

for all z E E. This monstrous formula occurs just like this on p.329 of ABEL's 1826 work [A].

4.

The R.iemann C-function. For all n E N , nz = exp(z log n) is

holomorphic in C and Inz l = nRz, according to the foregoing.

Theorem. The series Eio n-` converges uniformly in every half-plane {z E C : tz > 1 + C), e > 0 and converges normally in {z E C : Rz > 1}.

Proof. Fore > 0, In- 'I n3t` > n'+` if Iz > 1 + e. It follows from this that

00 1

n

^{1 < n e} for all z with Jtx 1 + e.

But it is well known that the series on the right converges whenever e > 0 (see, e.g., KNOPP [15], p. 115), so the theorem follows from the majorant criterion 3.2.2 and the definition of normal convergence.

The function

00

((z) RZ > 1

1 n

is therefore well defined and at least continuous in its domain of definition.

In 8.4.2 we will see that ((z) is actually holomorphic. Although EULER had already studied this function, today it is called the Riemann zeta-function.

At this point we cannot go more deeply into this famous function and its history, but in 11.3.1 we will determine the values ((2), ((4), ... , ((2n), .. .

Exercises

In the first three exercises G:= C \ {x E R : IxI > 1}.

Exercise 1. Find a function f E D(G) which satisfies f (0) = i and f 2(z) = z2 - 1 for all z E G. Hints. Set f, (z) := exp(2 log(z + 1)) for z E C \ {x E

R:x<-1} and f2(z):=exp(Zt(z-1)) forzEC\{xE R:x> 1}, where

e is an appropriate branch of the logarithm in C \ {x E R : x > 0}. Then consider fj(z)f2(z) for z E G.

Exercise 2. Show that q : C" -+ Cx given by z '-+ 1(z + z-1) maps the upper half-plane H biholomorphically onto the region G and determine the inverse mapping. Hints. In Exercise 3 to Chapter 2, §1 it was shown that q maps the upper half-plane bijectively onto G. Letting u : G -+ H denote the inverse mapping, show that u is related to the function f constructed in Exercise 1 above by

(*) (u(z) - z)2 = z2 - 1 = f2(z) , z E G.

Check that u is continuous and then infer from (*) that u(z) = z + f (z).

Exercise 3. Show that z '--+ cos z maps the strip S := {z E C : 0 < Rz < ir}

biholomorphically onto the region G. The inverse mapping arccos : G -+ S is given by

arccos(w) = -i log(w + w2 - 1)

where w - 1 suggestively denotes the function f (w) from Exercise 1 above.

Exercise 4. For G' C \ [-1,0] find a function g E O(G') such that

g(1)=fand g2(z)=z(z+1)for all zEG'.

Exercise 5. Determine the image G of G' := {z E C : 3tz < 3'z < Rz + 27r}

under the exponential mapping. Show that exp maps G' biholomorphically onto G. Finally, determine the values of the inverse mapping e : G - G' on the connected components of G fl R'.

Chapter 6

Complex Integral Calculus

Du kannst im GroBen nichts verrichten Und fangst es nun im Kleinen an (Nothing is brought about large-scale But is begun small-scale).

J. W. GOETHE

Calculus integralis est methodus, ex data differential- ium relation inveniendi relationem ipsarum quantita- tum (Integral calculus is the method for finding, from a given relation of differentials, the relation of the quan- tities themselves). L. EULER

1. GAUSS wrote to BESSEL on December 18, 1811: "What should we make

of f ipx dx for x = a + bi?

Obviously, if we're to proceed from clear concepts, we have to assume that x passes, via infinitely small increments (each of the form a+ i#), from that value at which the integral is supposed to be 0, to x = a + bi and that then all the cpx dx are summed up. In this way the meaning is made precise. But the progression of x values can take place in infinitely many ways: Just as we think of the realm of all real magnitudes as an infinite straight line, so we can envision the realm of all magnitudes, real and imaginary, as an infinite plane wherein every point which is determined by an abscissa a and an ordinate b represents as well the magnitude a + bi. The continuous passage from one value of x to another a + bi accordingly occurs along a curve and is consequently

167

possible in infinitely many ways. But I maintain that the integral f cpx dx computed via two different such passages always gets the same value as long as wx = oo never occurs in the region of the plane enclosed by the curves describing these two passages. This is a very beautiful theorem, whose not-so-difficult proof I will give when an appropriate occasion comes up.

It is closely related to other beautiful truths having to do with developing functions in series. The passage from point to point can always be carried out without ever touching one where cpx = oo. However, I demand that those points be avoided lest the original basic conception of f cpx dx lose its clarity and lead to contradictions. Moreover it is also clear from this how a function generated by f cpx dx could have several values for the same values of x, depending on whether a point where cpx = oo is gone around not at all, once, or several times. If, for example, we define log x via f =dx starting at x = 1, then arrive at logx having gone around the point x = 0 one or more times or not at all, every circuit adding the constant +27ri or -27ri; thus the fact that every number has multiple logarithms becomes quite clear." (Werke 8, 90-92).

This famous letter shows that already in 1811 GAUSS knew about con- tour integrals and the Cauchy integral theorem and had a clear notion of periods of integrals. Yet GAUSS did not publish his discoveries before 1831.

2. In this chapter the foundations of the theory of complex contour- integration are presented. We reduce such integrals to integrals along real intervals; alternatively, one could naturally define them by means of Rie- mann sums taken along paths. Complex contour integrals are introduced in two steps: First we will integrate over continuously differentiable paths, then integrals along piecewise continuously differentiable paths will be in- troduced (Section 1). The latter are adequate to all the needs of classical function theory.

In Section 3 criteria for the path independence of contour integrals will be derived; for star-shaped regions a particularly simple integrability criterion is found. The primary tool in these investigations is the Fundamental Theorem of the Differential and Integral Calculus on real intervals (cf.

0.2).

§0 Integration over real intervals

The theory of integration of real-valued continuous functions on real inter- vals should be known to the reader. We plan to carry this theory over to

complex-valued continuous functions, to the extent necessary for the needs of function theory. I = (a, b), with a < b will designate a compact interval

in R.