LANDAU and His Contemporary Critics. BALTZER'S method of introducing ir is not geometrical, but it is probably the most convenient way

ARGAND

6. LANDAU and His Contemporary Critics. BALTZER'S method of introducing ir is not geometrical, but it is probably the most convenient way

of arriving rapidly at in the real domain. Eduinund LANDAU (1877—1938) advocated and publicized this approach in his Ci6ttingen lectures and his EinfiV&rting in die Differentialrechnung und Iniegrizlrechnung (Verlag No-.

ordoff, Groningen) published in 1934, and written in his characteristic "tele- graphic" style. On page 193 of this book can be read "Die Welt konstanie aus Satz 262 werde dauernd mit bezeichneL" [The universal constant in Theorem 262 will always be denoted by LANDAU, who was a pupil of FROBENIUS, was appointed in 1909 Professor of Mathematics in Gcttingen as successor to MINKOWSKI. In 1933 he was dismissed on racial grounds.

There is an obituary notice by K. KNOPP in Jahresber. DMV, 54, 1951,

55—62.

The definition of 4ir as the smallest positive zero of cos z is now com- monplace. It is therefore all the more incomprehensible to us nowadays that this particular method of defining should have unleashed in 1934 an academic dispute for which the epithet "disgraceful" would be far too mild a description. A highly distinguished colleague in Berlin attacked LANDAU savagely. It will be enough to quote two of his sentences: "Uns Deutsche

lãBt eine soiche Rumpftheorie unbefriedigt" (Sonderausg. Sitz. Ber. Preuss.

Akad. Wiss., Phys.-Math. KI. XX, p. 6); und weitaus deutlicher: "So 1st die mannhafte Ablehnung, die em groBer Mathematiker, Edmund LAN- DAU, bei der Gôttinger Studentenachaft gefunden hat, letzten Endes darin begründet, daB der undeutsche Stil dieses Mannes in Forschung und Lehre deutsehem Empfinden unertraglich 1st. Em Yolk, das eingesehen hat,

...

wie Volk8fremde damn arbeiten, ihm fremde Art aufzuzwingen, muB Lehrer von einem ihm fremden Typus ablehnen." (Persönlichkeitsstruktur und math- ematisches Schaffen, Forach. u. Fortschr., 10. Jahrg. Nr. 18, 1934, p. 236.) [Such a tail-end of a theory leaves us Germans quite unsatisfied) and more specifically: [Thus

...

the valiant rejection by the Göttingen student body which a great mathematician, Edmund LANDAU, has experienced is due in the final analysis to the fact that the un-German style of this man in his research and teaching is unbearable to German feelings. A people who have perceived,

...

how members of another race are working to impose ideas foreign to its own must refuse teachers of an alien culture.]

Such abstruse, outrageous, and monstrous opinions were immediately and sharply rejected by the British mathematician G.H. HARDY in August 1934 in his note "The J-type and the S-type among the mathematicians"

(Collected Papers, 7, 1979,610—611) he wrote: "There are many of us, many Englishmen and many Germans, who said things during the War which we scarcely meant and are sorry to remember now. Anxiety for one's own position, dread of falling behind the rising torrent of folly, determination at all costs not to be outdone, may be natural if not particularly heroic excuses. Prof. Bieberbach's reputation excludes such explanations of his utterances; and I find myself driven to the more uncharitable conclusion that he really believes them true."

§2. THE EXPONENTIAL HOMOMORPHISM exp:C —' ^Cx

The exponential series, first written down for real arguments by NEWTON in a letter to LEIBNIz of the 24th October 1676 (see Math. Schriften, ed.

GERHARDT, vol. 1, p. 138)

Z2 Z3 °°

Zt'

2! 3! n!

is absoltttely convergentfor all z E C. This can be proved in exactly the same way as in the real case. We have thus defined in the whole of C a complex function exp: C —. C, which is called the (complex) exponential function, and which is the natural extension of the real exponential function into the complex domain. This function plays, ever since the days of EULER, a dominant role among the so-called "elementary transcendental functions."

We shall derive the Addition theorem which is of fundamental importance for the theory of the function exp z, from CAUCHY'S theorem on the product of two series:

Let b,, be absolutely convergent series. Then their "Cauchy product" PA, where p,, := is absolutely convergent, and

= EPA.

Thereader will find a proof of this theorem in thefamou8 Cours d'onalyse of CAUCHY, which appeared in Paris in 1821 (see, for example, Oeuvres,

3, Ser. 2, p. 237) and also in any modern Advanced Calculus text. The Addition theorem states that the mapping

exp:C_.CX,

_zi—.expz

is a homomorphism of the additive group C into the multiplicative group CX. Whenever a mathematician sees a group homomorphism o: C H, he immediately looks for the image group o(G), and the kernel

Kero := {g E C: o(g) = neutral element of H).

We shall show that, for the exponential homomorphism, exp(C) = ^CX, _{Ker(exp) =} 2,riZ,

where is a positive real number. To prove that exp(C) = ^CX we use a simple

Convergence Lemma. Corresponding to any w E there is a sequence

in CX with w and = 1.

We shall prove this straightaway. We write w = Iwic with c E S'. There is a c1 = ^a1 + it'1 E with 0 and c? =c. In view of the concluding remark in 3.3.5 we can now find a succession of numbers Cf% =

E S',

such that

=

0, *Ibn_iI. We see that ca" =^c and

It'll

Since > ¹ it

follows that limb, =

0, and therefore

=

lim(1 ^—

=

1,

so that =

¹ since

0. It is thus clear that

urnc,,

= +

^{= 1.} For the sequence defined by w, we

now have

=

wand

=

^1, because as is well known ¹

foranyr>O. ⁰

Apart from the convergence lemma, we shall also make use of two ele- mentary facts taken from the theory of functions (see J. Conway, Functions

of One Complez Variable, Springer-Verlag, 1978, p. 37).

1) Any power series f(z) = a,,z" is holomorphic inside its circle of convergence, and within this circle f'(z) =

2) 1ff is holomorphic and

f'

vanishes at every point inside some circle, then f is constant.

1. The Addition Theorem. (exp w)(exp z) =

exp(w+ z). To prove this we write

(expw)(expz) = ^with = _(A

_v)!v!WZ

A=O M+V=A

v0

by CAUCHY'S theorem on the multiplication of series. Now

1 1

1('A'\

(A_ri)!z4ARV)'

and consequently, by the Binomial theorem,

= =

÷

so that

A

(expw)(exp z) = ^(w =exp(w + z).

0

The Addition theorem asserts that the exponential function obeys the

"rule for powers." To bring out this point more clearly one often prefers to write

1 1

e2 :=expz, where

(EULER 1728).

If one uses this notation, which is not without its dangers, then the Addition theorem takes the suggestive form of the

Power Rule. =

for all w, z C.

If one puts w := —z in the Addition theorem, it follows from exp 0 = ^1,

that

(expz)' =exp(—z)

for all

ZEC;

and in particular that the function exp(z) has no zeros, so that it maps C into CX. _The Addition theorem now states that

The mapping exp:C — ^CX is a homomorphssm of the additive group C into the multiplicative group CX.

2. Elementary Consequences. The conjugation of convergent sequences is compatible with the formation of the limit. Consequently exp z = exp from which it follows that

(1)

Iexpzl=exp(Rez) forall zEC.

Proof. Since z + =

²Re z, we have, by the addition theorem

1 2 1 1 1

Iexpzi = exp

=

(exp

=

(exp

=exp(R.ez). ^ci

Since it is clear, from the form of the exponential series, that exp z > I

forx >

0,itfollowsthatexpx

=(exp(—z))1 <1 forx <0.Thestatement

(1) therefore implies (2)

and in particular y i— exp(iy) is a mapping of into the unit circle S'. As regards the behavior of its functional values we shall show that

(3) Im(exp(iy)) > 0 for 0 < y <

Proof. Since exp(iy) = and since R, we have Im(exp(iy)) ^{y —} + ^. .. +

—

(thesine series; cf. 3.1(2)). By writing this in the form

1 1 2"

we deduce at once that Im(exp(iy)) > 0for 0 < y < 0

Sinceexp(—iy) = we deduce directly from (3) the following.

Lemma. The only point in the open interval (—1, +1) at which the function

JR —.S', y exp(iy) has a real value is the pointy = 0.

The continuity of exp z is easily deduced from the Addition theorem.

Writing q =1 +1/2!+ 1/3!+•.. wehave lwlll+w/2!+

w2/3!

+ ..

.J < for all w C with Iwi 1. It follows therefore for

anycECandallzECwithiz—cl<1that:

—expci = Iexpcllexp(z^—c) — qiexpciiz — ci

and hence lexp z —expe' provided that iz —ci <min(1, Jqexp

0

It follows from the continuity of exp z with the help of the Intermediate value theorem, that, since exps> ¹

+ s for 8>

0 and exp(—z) = 1/expz, we have

(4)

expR={rEllt:r>O}.

3. Epiniorphism Theorem. The ezponential homomorphism exp C —i za an epimorphism, that is, ii is surjective.

Our proof is based on the following.

Lemma. There is a neighborhood U of the point 1 C such that U C exp(C).

Proof. The logarithmic series := z — + ^{— converges for}

izi < 1 and is therefore holomorphic with

=

^{1 —z+ z2 —z3}+ — =

(1 + z)'1 for Izi < 1, by the first statement in the Introduction. Similarly exp z ^is holomorphic in C, and (exp z)' =expz in C. Consequently 1(z)

(1 + z)exp(—)h(z)) also is holomorphic inside the unit circle, and it follows

by the chain rule that f'(z) is identically zero for all Izi < 1. Consequently, by statement 2 of the Introduction, 1(z) is a constant, and since 1(0) = ^1, we must have = ¹

+ z for all Izl < 1. It now directly follows

that the disc U := {z E C: Iz — < 1) lies in exp(C), because for any a E C satisfying a — < 1 the number 6 := )(a — 1) is well defined, and

expb=a.

⁰

The proof of the Epimorphism theorem can now be quickly completed.

By the convergence lemma in the introduction there exists, corresponding

to every w E CX, a sequence EC with =

wand

= 1. By the lemma in the preceding paragraph there exists also an index m 1 and a

I E C such that wm = exp I. For z

:= 2"I

we then have, by virtue of the Addition theorem, exp z= (exp I)2m = w,and we see that exp(C) = C. 0 We sketch a second proof of the Epimorphism theorem which works without the sequence For any WE the set of points W := {wz:z E

U} is a neighborhood of w in CX. If w E exp(C) then W c exp(C) by the lemma above and the group property of exp(C). Thus exp(C) is an open subgroup of the connected group CX. However, by an elementary theorem of the general theory of topological groups a connected group G has no open subgroups other than G.

In the proof of the above lemma the identity exp A(z) = 1 + z plays

the deci8ive role. It is possible to prove this identity in an elemenhzry way without use of the differential calculus. I am indebted to M. KNESER for the following argument. The proof is based on using the formulae:

(1)

= lim n[(1 +

— 11, z E F.,

(2)

expw = Jim (i

+ ^if =

where

ZEE.

From (1) and (2) may be deduced immediately, by taking := n[(1 +

— 1)

exp.X(z) = urn [(1

n—co

Thestatement asserted by the theorem now follows, if one also remembers that

(3) (1 + z)G(1 + ^{= (1}

+

^{z F..}

The statements (1)—(3) can be proved by elementary arguments, thus (3) is equivalent to identities involving binomial coefficients and

(a-I- b)

which hold for all natural numbers a, 6 and hence generally, by the binomial theorem.

In an appendix to this section we shall give another elementary proof of the lemma.