documenting riemann'scomplex
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8/11/2019 Documenting Riemann'SCOMPLEX
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E N E U E N S C H W N D E R
Docum enting R iem ann s
Im pact on tho Theory of
C om plex Functions
long w~th Euler Gauss and HzIbert Ber nhard R~emann ~s one of the mos t ~llus-
tmous m ath ema tic ian s of all t~me H~s pro min enc e ~n the fie ld of complex analy-
s~s may be appreciated s~mply by noting that ~n the
athemat~sches Worterbuch
by Naas-Schmld [1961, Vol 2,510-524], the ent nes relate d
to Rmmann's function-theoretical work (Rmmann map-
ping theorem, R mmann differential equation, Rmmann sur-
face, Rlemann-Roch theorem, Rmmann theta-functlon,
Rmmann sphere, Rmmann zeta-function, etc ) take up al-
most as much space as those related to all the work of
Euler or Gauss m total Rmmann's contnbu tlon to com-
plex analysis rests, on the one hand, on his publications,
especially his inaugural dissertation on the foundations of
complex analysis (1851) and his articles on the theory of
hypergeometnc and Abehan functions (1856-57), but his
several lecture courses in this field are also very impor-
tant
These were given in the yea rs fr om 1855 to 1862, and
regularly began with an introductor y general part Rmmann
then tur ned either to the theor y of elliptic and Abehan func-
tions or to hypergeometnc series and related transcen-
dental functions Much of the more advanced parts of
Rmmann's courses were published m his Collected Papers
[Rmmann 1990, 599-692] and m a boo k by H Stahl
[Rlemann 1899], but not his intro ducto ry lectures o n gen-
eral compl ex analysis Thus the latter gradually fell into
obhwon, despite their lntnnslc interest, and despite their
declsl've influence on later developments through the
closely related wntmgs of Dur~ge, Hankel, Koemgsberger,
Neumann, Prym, Roch, and Thomae It there fore seemed
appropnate to publish them m a cntmal edition
[Neuenschwander 1996], m order to make them accessible
to a broader circle of readers For this edition, I also pre-
pared an extensive bibliography of the history of the m~-
pact and influence of Rmmann's function theory
This newly assembled blbhography is mamly intended to
close---at least m the field of complex analysis--the con-
slderable gap existing between the blb hograplues of Purkert
and Neuenschwander, which were appended to the reprint
of Bernh ard R~eman n s Gesam melte Mathemat~sche Werke
[Rmmann 1990] The penod from 1892 to 1944, not system-
atically co~ered there, was scrutnuzed, usmg the indexes of
names m B~blwtheca Mathematzca (1887-1914), and the
Revue semestr~elle des Publ wat wns mathdmat~ques (1893-
1934/35), and ex ammmg the sections on Gescluchte und
Plulosoptue and Funktlonentheone (or Analysis ) m the
abstractmg journal Jahrbuch uber dze Fortschmtte der
Mathe~nahk (1868/71-1942/44) Additionally, I consulted
the holdings of older books in the libraries of the Institutes
of Mathematics at the Umversmes o f Gottmgen and Zurich
Generally, I included only publl cano ns which co ntmned ex-
plw~t reference to Rmmann's work (e g, quotations with
9 1 9 9 8 S P R I N G E R V E R L A G N E W Y O R K V O L U M E 2 0 N U M B E R 3 1 9 9 8 1 9
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8/11/2019 Documenting Riemann'SCOMPLEX
2/8
e x a c t i n d i c a t i o n o f l o c a t i o n ) , b u t e v e n t h i s c r l t e n o n l e f t
m o r e t h a n 1 ,0 00 o u t o f t h o s e 8 ,0 0 0 ti t l e s p r o v i s i o n a l l y s e -
l e c t e d o n t h e b a s i s o f t h e r e w e w s a n d t h e s y s t e m a t i c h -
b r a r y i n s p e c ti o n , t h e o n g m a l s w e r e t h e n l o o k e d u p , to
m a k e s u r e t h at t h e y m e t t h e c o n d m o n s f o r i n c lu s i o n
N a t u r a ll y , t h e n e w b l b h o g r a p h y Ls m n o w a y c o m p r e h e n -
s w e T h e h t e r a tu r e w [ u c h r e f e r s t o R m m a n n ' s p i o n e e n n g w o r k
i s a l m o s t b o u n d l e s s P u r k e r t , fo r e x a m p l e , e x a m i n i n g a b o u t
t e n j o u r n a l s f r o m t h e f i r st 2 5 y e a r s a f t e r t h e d e a t h o f B e r n h a r d
R i e m a n n , a l r e a d y f o u n d m o r e t h a n 5 00 p u b h c a t m n s r e f e r ri n g
t o t u s w o r k A c c o r d i n g to d a t a b a s e s u r v ey s , t h e c o n t i n u a t i o n
a n d e x t e n s i o n o f P u r k e r t ' s re s e a r c h u p t o t h e p r e s e n t w o u l d
h a v e t o t a k e i n t o a c c o u n t a b o u t 3 0, 00 0 p u b h c a t m n s , a l l o f
w [ u c h o b w o u s l y c a n n o t b e e x a m m e d m c hv ~ du al ly w i t h i n a r e a -
s o n a b l e t u n e N e v e r t h e le s s , I h o p e t h a t t h e n e w b l b h o g r a p h y
w i l l b e c o m e a u s e f u l t o o l f o r f u r t h e r m v e s U g a t ~ o ns
I n t h e f o l l o w i n g I w i l l t r y t o i l l u s t r a t e s o m e o f i t s p o s -
s i b l e a p p li c a t i o n s , b y s u r v e y m g t h e i m p a c t o f R i e m a n n ' s
f u n c t i o n - t h e o r e t i c a l w o r k m t h e f o u r m o s t i m p o r t a n t
E u r o p e a n c o u n t r i e s G e r m a n y , F r a n c e , I t al y , a n d G r e a t
B n t m n S p e c i a l a t t e n t i o n w i ll b e g i v en t o th e d e v e l o p m e n t s
i n G r e a t B n t m n b e c a u s e t h e y h a v e n o t y et b e e n a n a l y s e d
i n d e t a i l F o r m o r e s p e c i f i c m f o r m a U o n , t h e r e a d e r m a y
t u r n t o t h e b i b l i o g r a p h y i ts e l f
G e r m a n y : E a r l y a n d S u s t a i n e d R e c e p t i o n o f
R i e m a n n s M e t h o d s
F o r a p r e h m m a r y n n p r e s s l o n o f t h e s i t u at i on m G e r m a n y , l e t
u s f i rs t l o o k a t t h e r e f e r e n c e s t o R m m a n n ' s w o r k m A u g u s t
L e o p o l d C r e l l e 's m f l u e n t l a l Journal fu r d~e re,he u nd ange-
wandte Mathemat~k I t ~s n o t e w o r t h y t h a t t h e f i r st o f t h e s e
r e f e r e n c e s g o e s b a c k t o H e l m h o l tz [Crelle 5 5 ( 1 8 5 8 ) , 2 5 - 5 5 ] ,
w h o , a s w e k n o w , l a t e r d i s c u s s e d R l e m a n n ' s h y p o t h e s e s
c o n c e r m n g t h e f o u n d a t io n s o f g e o m e t r y H e l m h o l t z w a s
f o l l o w e d i n c h r o n o l o g i c a l o r d e r b y l _ a p s c [ u t z , C l e b s c h ,
C h n s t o f f e l , S c h w a r z , B n l l , F u c h s , G o r d a n , L u r o t h , a n d
W e b e r , a l l o f w h o m , e v e n t h o u g h t h e y w e r e n o t R m m a I m ' s
i m m e d i a t e p u p i ls , d i d a g r e a t d e a l t o d ~ s s e m i n a t e tu s i d e a s ,
a s d i d [ u s o w n s t u d e n t s R o c h , T h o m a e , a n d P r y m
[Crelle
6 1 ( 1 8 6 3 ) - 7 0 ( 18 6 9) ] C l e b s c h a n d B r l l l - - h k e K l e in a n d
N o e t h e r - - p u b h s h e d t h e i r la t e r w o r k p r i m a r i l y in t h e
Mathemat~sche Annalen , w h i c h s t a r t e d t o a p p e a r m 1 86 9,
t h u s , c o n s u l t i n g C r e l l e ' s Journal a l o n e p r o w d e s o n l y m -
c o m p l e t e r e s u l t s f o r t h e m A c c o r d i n g t o o u r b i b l i o g r a p h y ,
o t h e r i m p o r t a n t e a r ly p r o m o t e r s o f R m m a n n 's i d e a s m t h e
G e r m a n - s p e a k i n g w o r l d w e r e C a n t o r , D e d e k m d , D u B o ls -
R e y m o n d , D u r ~ g e, H a n k e l, K o e m g s b e r g e r , N e u m a n n ,
S c h l a f h , a n d , m [ u s l a t e r y e a r s , S c h o t t k y i
I ta l y : E n t h u s i a s t i c A p p r e c i a t i o n o f R i e m a n n s W o r k
A n ~ m p r e ss i ve p m t u r e o f th e e x t e n t t o w [ u c h R m m a n n ' s w o r k
w a s a p p r e c i a t e d m I t a l y i s g w e n b y a s i m i l a r s t u d y o f t h e r e f-
e r e n c e s t o [ u s w r i t i n g s m t h e Annal~ dz Matematwa pura
ed apphcata,
a j o u r n a l w h i c h p l a y e d a n o u t s t a n d i n g r o l e m
t h e d m s e m m a t i o n o f R i e m a n n ' s t h o u g h t s A s e a r ly a s 18 59 ,
E n r i c o B e t t l, w h o w a s t o b e c o m e R i e m a n n ' s f r i en d , t r a n s -
l a t ed [ u s & s s e r t a t m n [Annalz 2 ( 1 8 5 9 ) , 2 8 8 - 3 0 4 , 3 3 7 - 3 5 6 ] , t o
w h i c h h e s o o n r e t u r n e d m a n e x t e n s i v e a r t i c le o n th e t h e -
o r y o f e l l i p t i c f u n c t i o n s [Annah 3 ( 1 8 6 0 ) , 6 5 - - 1 5 9 , 2 9 8 - 3 1 0 ,
4 ( 1 86 1 ), 2 6 - 4 5 , 5 7 - 7 0 , 29 7 - 3 3 6 ] I n t h e s a m e v o l u m e s , w e
a l s o f in d a r e p o r t b y B e t ti o n R i e m a n n ' s t r e a t i s e c o n c e r n -
m g t h e p r o p a g a t i o n o f p la n a r m r w a v e s
[Annah
3 ( 1 8 6 0 ) ,
2 3 2 -2 4 1 ], a n d a r e p o r t b y A n g e l o G e n o c c [ u o n R i e m a n n ' s
i n v e s t i g a ti o n o f th e n u m b e r o f p r i m e s l e s s t h a n a g i v e n
b o u n d
[Annals
3 ( 1 8 6 0 ), 5 2 - 5 9 ] I n a l a t e r v o l u m e o f t h e
s a m e j o u r n a l [Annalz ( 2 ) 3 ( 1 8 6 9 - 7 0 ) , 3 0 9 - 3 2 6 ] , w e f i n d a
F r e n c h t r a n s l a t i o n o f R m m a n n ' s h y p o t h e s e s o n t h e fo u n -
d a t io n s o f g e o m e t r y b y t h e F r e n c h m a t h e m a t i c i a n J u l e s
H o u e l, w h o t y p i c a l l y d i d n o t p u b h s h [ u s t r a n s l a t m n i n a
F r e n c h j o u r n a l , b u t i n t heAnnah W e s h o u l d a l s o m e n t i o n
E u g e m o B e l t r a m l a n d F e h c e C a s o r a t i , t h e l a t t e r h a v m g a l -
r e a d y p r e s e n t e d R m m a n n ' s t h e o r i e s m 1 86 8 m a b o o k
[ C a s o r a t i 1 86 8] a n d m s p e c i a l l e c t u r e s g i v e n m M i l a n
[ A r m e n a n t e & J u n g 1 86 9, C a s o r a U & C r e m o n a 1 86 9] I n w e w
o f t[ u s e x c e l l e n t m t r o d u c t i o n a n d t r m l - b l a zm g , i t i s n o t s u r -
p n s m g t h a t R m m a n n ' s t h e o r i e s w e r e w i d e l y k n o w n m I t a ly ,
a n d t h a t t h e y w e r e q u o t e d m m o r e t h a n t h u - ty a r t i c l e s m t h e
Annalz a l o n e u p t o 1 89 0 2 A s t o R m m a n n ' s o w n s t a y s m I t a ly ,
a n d o t h e r f o l l o w e r s o f R I e m a n n t h e r e , s e e t h e a r t i c l e s o f
B o t t a z z n u , D m u d o n n ( , L e n a , N e u e n s c h w a n d e r , S c h e r m g ,
T n c o n u , V o l t e r r a , a n d W e I1 c i t e d m t h e b i b l i o g r a p h y
F r a n c e : H e s i t a n t R e c e p t i o n
In France, the s i tualaon was ra hcaUy di f ferent Skmmung
t h r o u g h t h e p a g e s o f th e Journal de Mathdmat~ques pures et
apphqudes e & t e d b y J o s e p h L l o uv l ll e , o n e f i n d s a l m o s t n o r e f-
e r e n c e s t o R ] e m a n n ' s p a p e r s b e f o r e 1 87 8, a n d m o t h e r F r e n c h
p u b h c a t~ o n s u p t o 1 88 0 th e y a l s o s e e m t o b e r e l a a v e l y s p a r s e
F u r t h e r m o r e , a c e r t m n c n t ] c a l r e s e r v e a s t o t h e u s e f u l n e s s
o f R m m a n n ' s m e t h o d s q u i t e o f te n c o m e s t h r o u g h B n o t a n d
B o u q u e t, f o r i n s t a n c e , w r i t e i n t h e f o r e w o r d t o th e s e c o n d
e d i t i o n o f t h e i r Thdor~e des fo nc tw ns ell~pt~ques [ B n o t &
B o u q u e t 1 8 75 , I f ]
In Cauchy s theory, the path of the ~mag~nary [complex]
vartable ~s characterized by the movement of a point on
a plane To represent thosefunc tzons whzch assume sev-
eral values fo r the same value of the vamable, Rzem ann
regarded the plane as for med of several sheets, sup emm-
posed and welded together, ~n order to aUow the vartable
to pas s fro m one sheet to another, wh~le t~avers~ng a con-
nect~ng [branch] hn e The concept of a many-s heet ed sur-
fac e presen ts some d~ffienlt~es, despzte the beautiful re-
sults whwh R~emann achieved by th~s method, ~t dzd not
seem to us to offer any adv antage fo r our own ob2ectwe
Cauchy s zdea ~s very suitable for representing multi-
valued fu nc tw ns , ~t ~s suffw~ ent to 2ozn to the value of
the variable the corresponding value of the functzon, and
1For b~bl~ographrcal detatls on part icular ar t icles by these authors c~t~ng R~emann see [Neuenschwander 199 6 131 2 32]
2The authors o f other ar ttc les in the nnah containing references to R~emann are Asco h Beltramt C asoratl Cesaro Chnstoffe l Dtn l Ltpsch~tz Pascal Schlafh Schwarz
Tonelh Volterra etc For detai ls see [Neuenschwander 1996]
2
THE MATHEMATICAL NTELLIGENCER
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8/11/2019 Documenting Riemann'SCOMPLEX
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w h e n t h e v a ~ a b l e h a s d e s c r i b e d a c l o s e d
c u r v e a n d t h e v a l u e o f t h e f u n c t w n h a s
t h e r e b y c h a n g e d , t o ~n d z c a te t h z s c h a n g e b y
a n ~ n d e x
O n l y H o u e l, m e n t m n e d a b o v e , t r i e d q ui te
e a r l y t o p r o m o t e R l e m a n n s ~ d ea s, a n d , t o -
g e t h e r w i t h G a s t o n D a r b o u x , h e s e v e r a l U m e s
d e p l o r e d t h e i r n o t b e i n g b e t t e r k n o w n m
F r a n c e [ G l s p e r t 1 9 8 5 , 3 8 6 - 3 9 0 e t p a s s i m ] A
r e w e w o f H o u e l s p m n e e n n g b o o k m t h i s f ie ld
[ H o u e l 1 8 6 7 - 1 8 7 4 , p a r t s 1 a n d 2 (1 8 6 7 / 6 8 ) ] c o n -
c l u d e s w i t h t h e f o l l o w i n g , q u i t e m s t r u c t w e
p a s s a g e [ N o u ve l le s A n n a l e s d e M a t h e m a -
t ~ q u e s ( 2 ) 8 ( 1 8 6 9 ) , 1 3 6 - 1 4 3 ]
M a y t h e ~ e c e p t w n o f t h~ s w o r k e n c o u r a g e t h e
a u t h o ~ t o k e e p h ~s p r o m ~ s e t o g ~ v e u s, z n a t h u d
p ar t~ a n e x p o s z t w n o f R z e m a n n s t h eo r i es ,
w h i c h u p to n o w h a v e b e en a l m o s t u n k n o w n
~ n o u t c o u n t r y , a n d w h w h o u t n e i g h b o u r s c u l-
t w a t e u u t h s o m u c h a r d o u r a n d s u c c e s s ~
@
lO
I n s p i t e o f H o u e l s e f f o r t s , t h e s i t u a t i o n m ~ s ~
rance seems to have changed fundamentally
o n l y a f t e r 1 88 0, w h e n a n e w g e n e r a t i o n o f
F r e n c h m a t h e m a h c m n s t o o k o v e r, t h e m o s t / ~ /
p r o m i n e n t a m o n g t h e m w e r e H e n r i P o m c a r e ~
( 1 8 5 4 - 1 9 1 2 ) , P a u l A p p e l l (1 8 5 5 - 1 9 3 0 ) , E m i l e
P l c a r d ( 1 8 5 6 - 1 9 4 1 ), a n d l ~ d o u a r d G o u r s a t
( 1 8 5 8- 1 9 3 6) W i t h t h e s e a u t h o r s t h e r e c e p t m n
o f th e w o r k s o f R m m a n n s f o l lo w e r s ( m m n l y
C l e b s c h a n d F u c h s ) n o w b e g a n t o l e a d t o a
m o r e t h o r o u g h r e a d m g o f R m m a n n s o w n w n t -
m g s - - a s n u m e r o u s q u o t a t i o n s m th e i r p u b h c a -
t m n s p r o v e A s a m a m f e s t a t ] o n o f t h e m c r e a s -
m g a p p r e c l a O o n o f R m m a n n s m e t h o d s a m o n g
F r e n c h m a t h e m a t m m n s a t t h a t t u n e, w e m a y
t a k e t h e f a c t th a t , m 1 88 2, G e o r g e s S n n a r t w r o t e
a d i s s e r t a t a o n o f 1 23 p a g e s , n o w a l m o s t c o m -
p l e t e l y f o r g o t t e n , m w h i c h h e t r i e d t o a c q u a i n t F r e n c h m a t h -
e m a t m m n s w ~ t h R m m a n n s f u n c t m n - t h e o r e t~ c a l tr e a t i s e s I n
S L m a r t s m t r o d u e t m n , w e f i n d th e f o l l o w m g s t a t e m e n t , w t u c h
c o n f i r m s m y o w n a s s e s s m e n t [ S u n a r t 1 88 2, 1 ]
~
d c . / 4 Y 3 7
g z/ / ,Z. /7 3 J-
/ 1~ 3 . , / -,4- d 7
F , g u r e 1 D , s c u s s , o n a n d r e p r e s e n t a t i o n o f a b r a n c h p o i n t o f a n a l g e b r a i c f u n c t , o n
o n a R l e m a n n s u r f a ce , t a k e n f r o m R m m a n n s l e c t u r e s ,n th e s u m m e r s e m e s t e r o f
1 8 61 T h e r e p r o d u c e d p a g e c o m e s f r o m a s e t o f l e c t u re n o t e s m a d e b y E d u a rd
S c h u l t z e w , t h a n n o t a t i o n s b y H e r m a n n A m a n d u s S c h w a r z , , t , s k e p t , n t h e S c h w a r z
N a c h l a s s a t th e A r c h w d e r B e r | , n - B r a n d e n b u r g , sc h e n A k a d e m , e d e r W , s s e n s c h a f t e n
, n B e r l ,n T h e r e e x , s t s a s , m , l a r r e p r e s e n t a t m n m t h e ( p r o b a b l y ) o r , g , n a l n o t e s b y
S c h u l t z e w h , c h a r e n o w k e p t m t h e R a r e B o o k a n d M a n u s c r , p t L i b r a ry a t C o l u m b , a
U n , v e r s lt y , n N e w Y o r k B o t h m a n u s c r , p t s w e r e d , s c o v e r e d b y t h e a u t h o r , i n 1 9 79
a n d 1 9 8 6 , r e s p e c t , v e l y F o r f u r t h e r , n f o r m a t , o n a n d a t r a n s c r , p b o n o f t h e t e x t , s e e
N e u e n s c h w a n d e r 1 9 8 8 a n d 1 9 9 6 , p p 7 5 f a n d 84 , M S S a n d
s
M e m o i r s ~ eq u~ re s a k n o w le d g e o f R w m a n n ~ a n s u r f a ce s
[ a n d t h u s, f ~ n a U y , o f R ~ e m a n n s o w n w r t t ~ n g s ] , t h e u s e
o f w h i c h h a s b e c o m e s t a n d a r d a t c e r t a i n G e r m a n u r n -
w e t s 1 ~ e s
W e k n o w t he m a g m f i c e n t r e s ul ts w h z e h R z e m a n n a r r t v e d
a t ~ n h z s t w o M e m o i r s c o n c e r n z n g t h e g e n e r a l th eo ~? ] o f
a n a l y tz e f u n c t w n s a n d t he t he o r q o f A b e h a n f u n c t w n s ,
b u t th e m e t h o d s w h i c h h e u s e d - - a n d e x p l a i n e d p e r h a p s
t oo s u e c ~ n e t l y - -- a ~ e n o t w e l l k n o w n ~ n F ~ a n c e [ ] A t
t h e s a m e t z m e , h o w e v e r , R ~ e m a n n s m e t h o d c o n t i n u e s t o
h a v e m a n y a d h e r e n t s ~ n G e r m a n y , ~ t s e r v e s a s a ba s ~s
f o r m a n y z m p o r ta n t p u b h c a t w n s b y f a m o u s g e om e t er s ,
s u c h a s K o e m g s b e r g e r , C a r l N e w m a n n [ s~ c], K l e z n ,
D e d e k z n d , W e b e r , P r y m , F u e h s , e t c T h e r e a d i n g o f t h e s e
G r e a t B r it a in A F o r g o t t e n I n t e r e s t
I n c o n t r a s t to w h a t i s k n o w n a b o u t t h e r e c e p h o n o f
R m m a n n s t h e o r i e s m t h e t h r e e c o u n t r i e s a l r e a d y s u r v e y e d ,
t h e i r r e c e p t i o n m G r e a t B n t m n i s h t t l e s tu d m d L e t m e
t h e r e f o r e p r o v i d e a m o r e d e t m l e d a c c o u n t
I n G r e at B n t m n R m m a n n s t h e o r i es s e e m t o h a v e b e -
c o m e k n o w n a n d m o r e w i d e ly d i s s e m i n a t e d o n l y a f te r h is
d e a t h O n e o f t h e fi r st B n t m h m a t h e m a t i c i a n s w h o c i t e d
R m m a n n f r e q u e n t l y w a s A r t h u r C a y l e y , w h o , a l r e a d y m
1 8 65 /6 6 , m e n t i o n e d t h e i n v e s t i g a t i o n s o f C l e b s c h a n d
3Other scattered early ctta ons of R~emann s wnttngs are to be found tn Bertrand Darboux Elhot Emmanuel Herm~te Jonqu~eres Jordan Mane and Tannery For
further informa n see [Neuenschwander 1996]
VOLUME 2 NUMBER 3 1998 21
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t m l a n d w i t h a p p r e c m t m n T h i s c a n b e s e e n f r o m h is n o t e
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t h e A d v a n c e m e n t o f S c i e n c e ( 1 8 83 ) O t h e r e a r l y c ] t a t m n s
o f R m m a n n ' s p a p e r s m G r e a t B r i ta i n a r e b y W T h o m s o n
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m e m o i r o f B e r n h a r d R ~ e m a n n , U e b er d ~e A n z a h l d e~
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t h e c e l eb r a te d m e m o z r , S u r l e s N o t a b l e s t b e m ~ e ~ s , b y
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t a b l e s w e r e c i t e d m t h e t r t tu ] l b y v a r i o u s m a t h -
e m a t m t a n s m S c a n dm a v ~ a n c o u n t r i e s ( O p p e r -
m a n n 1 88 2 f , G r a m 1 8 84 f f , L o r e n z 1 8 9 1) ,
w h e r e R m m a n n ' s f u n c t m n - t h e o r et m a l w o r k
h a d a l r e a d y e x c i t e d i n t e r e s t b e f o r e , a s i s
s h o w n b y t h e p u b l i c a t i o n s o f B o n s d o rf f ,
M l t t a g - L e ff l e r , a n d s o m e o t h e r s M o r e o v e r , m
1 88 4 W W J o h n s o n p u b h s h e d a d e t a i l e d a c -
c o u n t o f J a m e s G l a l s h e r ' s f a c t o r t a b l e s a n d
t h e d l s t n b u t m n o f p r i m e s m t h e A m e r i c a n
j o u r n a l Annals of Mathemahcs w h i c h m t r o -
d u c e d R m m a n n ' s m v e s t i g a t m n s a n d t h e w o r k
o f t h e B r i t is h m a t h e m a t l c t a n s m t h e N e w
W o r l d
F i n a l l y , a t t e n t i o n s h o u l d b e p r o d t o
A n d r e w R u s s e l l F o r s y t h , w h o w e n t t o l i v e m
t h e t o w n o f C a m b r i d g e a n d s t u d i e d n o t h i ng
b u t m a t h e m a t m s m t h e s a m e y e a r , 1 87 6, t h a t
S m i t h a n d G l m s h e r s t a r t e d t o d r a w a t t e n t m n
t o R l e m a n n ' s w o r k I n h i s o b i t u a r y n o t i c e,
E T W h i t t a k e r d e s c r i b e d F o r s y t h ' s Theory of
Functwns ( 1 8 93 ) a s h a w n g h a d a g r e a t e r i n -
f l u e n c e o n B r it i s h m a t h e m a t i c s t h a n a n y w o r k s i n c e
N e w t o n ' s Pnnc~p a A c c o r d i n g t o W h i t t a k e r [ 19 42 , 21 8 ],
p e r h a p s t h e m o s t o r i g i n a l fe a t u r e o f th ~s w o r k w a s t h e
m e l d i n g o f t h e t h r e e m e t h o d s a s s o c m t e d w i t h t h e n a m e s
o f C a u c h y , R m m a n n , a n d W e m r s t r a s s , w h i c h m t h e c o n t i-
n e n t al b o o k s w e r e r e g a r d e d a s s e p a r a t e b r a n c h e s o f m a t h -
e m a t l c s F u r t h e r m o r e , o n e c a n r e a d m t h e R o y a l S o c m t y
B m g r a p h l c a l M e m o i r o f H e n r y F r e d e r i c k B a k e r b y W V D
H o d g e t h at F o r s y t h r e n d e re d C a m b n d g e m a t h e m a t i c s
g r e a t s e r v i c e b y h i s e f f o rt s to b n n g a b o u t c l o s e r c o - o p e r a -
t i o n b e t w e e n m a t h e m a t ~ c i a n s m t h is c o u n t r y a n d t h o s e o n
t h e c o n t i n e n t o f E u r o p e , a n d h e m a d e i t e a s m r f o r h~ s
5In add i t i on t o Sm i t h s Lon don a ddres s d~ scussed a bove see a l so h~s pap er
On the ~ntegrat~on of d~scont~nuous functions
[1875) w=th a deta= led analys~s of R~emann s
m e m o i r o n t h e r e p r e s e n t a ti o n o f a f u n c t io n b y a t n g o n o m e t n c s e n e s h ~s p a p e r On some discontinuous senes considere d by R~ernann 1881) and h~s M e m o i r o n t h e
Theta and Omega unc tsons w h i c h w a s w n t t e n t o a c c o m p a n y t h e T a b l e s o f t h e T h e t a F u n c t io n s c a l c u l a t e d b y J W L G la ~ sh e r
JOLUME 20 NUMBER 3 199 8 2 3
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successors, mclu(hng Baker, to get the full benefit of the
work of the great German masters of the late nmeteenth
century
As can be seen from W[uttaker's obituary notice and
Forsyth's own recollections of his undergraduate days
[Forsyth 1935], there is a defimte link betwee n For syt h's
later work m complex analysis and [us early studies in
Cambridge, where he came mto repeated contact w~th
Glmsher, Cayley, and the works of Smith, who taught at
Oxford Cayley and Glmsher were also to be very instru-
mental in Forsyth's caree r in pure mathematic s and helped
[urn to overco me that Cambridge atmos phere m which
all were reared to graduation on a pphed m athemat ics ,,6
Accorchng to Whittaker, it was Glmsher who suggested to
Forsyth that he write his fLrst book, the
T r e a t i s e o n d ~ f -
f e r en t ~ a l e q u a t w n s
(1885) His first two princ ipal memo irs
on the ta functions (1881/1883) and on Abel's the orem and
Abehan functions (1882/1884) were presented by Cayley,
on the other hand, to the Royal Society In the first mem-
oir Forsyth gives a list of 22 prmcipal papers in the field,
mcluchng among others Rmmann's T h e o r t e d e r A b e l s c h e n
F u n c t w n e n a s well as twelve other papers by the German
mathe mati cian s Jacobi , Rlchelot, Rosenhmn, Gopel,
Wemrstrass, Koemgsberger, Kummer, Borchardt, and
Weber The second memoir on Abel's theorem and Abeha n
functions contained a large section on Wemrstrass's ap-
proach, which clearly shows tha t around 1882 Forsyth was
already fully aware of the achievements of the German
mathematicians in this field
o n c l u s i o n
These bibhograplucal mvestlgaUons make it evident that
Rmmann's ideas were m ore posltavely acce pted m Great
Britain, apparently even earher, than m France It seems the y
were among the more important stimuh wluch, through the
efforts of Cayley, Clifford, Slmth, Glmsher, etc, later led to
the flounshmg of Enghsh pure mathematacs and function
theory under Hobson, Forsyth, Mathews, Baker, Barnes,
Hardy, IAttlewood, T~tchmarsh, and many othe rs In England
and Italy Rmmann's theories entered open temtorie s, where
they found mathematmal commumtms eager to catch up m
pure math ematics In England this interest serve d to
broad en to the Cambridge atmosphere, which was nearly
totally oriented towards natural philosophy and apphed
mathematics, m Italy it fitted m w~th a desire to built up a
modern mathematical education for the newly founded na-
tion (cf N euensc hwander 1986) In France, on the other
hand, Riemann' s ideas first faced a cold receptaon from the
well-esta bhshed tradition of Cauchy and Briot & Bouquet
Allowing for a certmn p olitical ly motiv ated enthus iasm,
Welerstrass was therefore proba bly right when, m a letter
to Casorati dated 25 March 1867 (written m the early pe-
riod before the publication of Rmmann's Collected
Mathemat ical Works m 1876), he espec ially empha sized
Italy's role m the dissemination of Rmmann's ideas In that
letter (see [Neuen schwan der 1978, 72]) 7 he writes
T h e h a p p y r ~ se o f s c w n c e ~n y o u r f a t h e r l a n d w a ll n o w h e r e
e l se b e f o l l o w e d w z t h m o r e v z v z d z n t e r e s t t h a n h e r e z n
N o r t h G e r m a n y a n d y o u m a y b e c e r t a i n t h a t t h e State
o f I t a l y h a s n o w h e r e e l se s o m a n y s ~ n c e r e a n d d ~ s~ n te r -
e s t e d f r i e n d s W z t h pl ec a~ u re w e a r e t h e r e f o r e r e a d y t o co n -
t ~ n u e t h e a l l i a n c e b e t w e e n y o u a n d u s - - w h w h ~ n t h e p o -
l ~ t w a l f i e l d h a s h a d s u c h g o o d r e s u l t s - - -a l s o ~ n s c w n c e s o
t h a t a ls o ~n t h a t a~ e a th e b a r r i e r s m a y m o r e a n d m o r e b e
o v e rc o m e w ~ t h w h w h u n h a p p y p o l i t i c s h a s f o r s o lo n g s e p-
a r a t ed t w o g e n e r a l ly c o n g en i a l n a t w n s T h e p a pe ~ w h z c h
y o u s e n t t o m e p r o v e s t o m e o n c e a g a i n t h a t o u r s c i e n t i f i c
e n d e a v o u r s a r e b e t te r u n d e r s t o o d a n d a p p r e c z a t e d ~ n I t a l y
t h a n ~ n F ~ a n c e a n d E n g l a n d p a r t z c u l a r l y ~ n t h e l a t t e r
c o u n t r y w h e r e a n o v e r w h e l m i n g f o r m a h s m t h r e a te n s t o
s t i f l e a n y f e e l z n g f o r d e e p e r ~ n ve s t~ g a t~ o n H o w s ~ g n ~ f i ca n t
~ t ~s t h a t o u r R w m a n n w h o s e lo s s w e c a n n o t s u f f i c i e n t l y
d e p lo r e ~ s s t u d ~ e d a n d h o n o u r e d o u t s i d e G e r m a n y o n l y
~ n I t a l y ~ n F r a n c e h e ~s c e r t a i n l y a c k n o w l e d g e d e x t e r n a l l y
b u t l~ t tl e u n d e r s t o o d a n d ~ n E n g l a n d [ a t l e a s t b e f o r e 1 8 6 7 ]
h e h a s r e m a i n e d a l m o s t u n k n o w n
N o t e a d d e d t n p r o o f
In a recent issue of this journal
(vol 19, no 4, Fall 1997) there app ear ed an article by
Jeremy Gray w[uch gives the r eader a very welcome sum-
mary m Enghsh of Riemaim's introduct ory lectures on gen-
eral complex analysis Gray's pape r is largely based on my
preprmt
R ~ e m a n n s V o r le s u ng e n z u r F u n k t ~ o n e n th e o m e
A l l g e m e ~ n e r T e ~ l
Darmstadt 1987 and on Roch 1863/65
(cf Neue nsch wand er 1996, p 15) but does not mentio n the
substantially expande d version which app eared m Sprmg
1996 and which has been reviewed m MR 97d 01041 and
Zbl 844 01020 The publis hed version also contams, be-
sides what was m the original prepnnt, the above-men-
tioned extensive bibliography of papers and treatis es that
were influenced by Rmmann's work, as well as a list of all
known lecture notes of [us courses on complex analysis
From this list [Ne uensc hwande r 1996, 81 ff ], one can in-
fer that Cod Ms Rmmann 37 comp rise s notes of three dif-
ferent courses (not just one, as suggeste d by Gray), and its
first part is thereby quite easily datable to the Winter se-
me st er 1855/56 On page 111 of Cod Ms Rmmann 37 one
6Accordtng to Forsyth s own recol lecttons as ear ly as h~s s tude nt years he to ok an exceptional ~nterest tn pure mathematics and w ent for one term in hts th ird year to
Cayle y s lectures whe re at the beginning the very wo rd plunge d him ~nto complete bewilderment From othe r passages of Forsyth s recol lections o ne may see that
already at that t~me he made h~s f i rs t t imid ventures outs ide the range of Cambridge textbo oks and plo ugh ed throug h amo ng others a large part of Durege s
lhpt~sche
Funct~onen
Further detai ls ma y be gathered from the fo l lowing personal confess ion by Forsyth Som ething of d~fferentta l equations be yon d mere examples the ele
mer i ts of Jacobean elhpttc functtons and the mathem attcs o f Gauss s metho d of least squares I had learnt from Glatsher s lectures and b y worktng at the matter of
the one course by Cayley whtch had be en attended I bega n to unde rstand that pure mathemattcs w as mo re than a col lectton of random to ols matnly fashioned for
use ~n the Cam bndg e treatment of natural phtlosophy Otherwtse ve ry near ly the whole of such k now ledge of pure m athematics as ~s m~ne bega n to be acquired only
a f te r my Tnpos degree In tha t Cambn dge a tmosphere we
ll
were reared to graduation on appl ied mathematics [Forsyth 1935 172]
7Nottce that h~s charactertzatton of th e s~tuat~on ~n England ts s tr tk tngly s tm~lar to th at o f Smtth see note 4) o r Forsyth see note 6)
2 4 THE MATHEMATICAL NTELLIGENCER
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A U T H O I
ERWIN NEUENSCHW NDER
Instrtut fur Mathemattk
Abt Re~ne Mathemabk
Un~vers~tat Zunch-lrchel
Wtnterthurerstrasss 190
8057 Zunch Switzerland
Erw~n Neuenschwander ts Professor of the History of
Mathematics at the University of Zurich He took his degrees
under B L van der Waerden ~n Zurich He has published sev-
eral papers on ancient, medieval, and n~neteenth-century
mathematics, an d especially on Rtemann, to wh om h~s latest
boo k ~s devoted Dunng the pest cou ple of years, he has also
take n an ~nterest ~n he h~story of the othe r exact so|en ces and
on th e ~nteracttons between science, philosophy, and society
The p~cture shows h~m before the tom b of Rtemann at Selesca
near Verban~a, Italy) on the occes~on of t he Verban~a R~emann
Memo nal Co nferen ce ~n July 1991
reads Fortsetzung ;m Somme rsem este r 1856 [Continua-
tion m the Summe r semeste r of 1856], and on page 193 of
the same manuscript Theone der Functmnen complexer
Groigen mlt besonderer Anwendung auf die Gauf~'sche
Re;he
F a,/3,
% x) und verwandte Transcendenten Winter-
semester 1856/57 Furthermore, ;t should be noted that
Rmmann's lectures on general co mplex analys;s developed
gradually from 1855 to 1861 [Neuenschwander 1996, 11 f ],
and that m 1861 he also treated We|erstrass's fac tonzat mn
theorem for enUre functions w;th prescribed zeros, intro-
ducing to th;s end logarithms as convergence-producing
terms [Neue nschwa nder 1996, 62-65], which ;s quite re-
markable, although not mentioned by Gray 1997
c k n o w l e d g m e n t s
I am indebted to Ro bert B Burckel (Manhattan, Kansas),
Ivor Grattan-Gumness (Enfield), and Samuel J Patter son
(Gottmgen) for various suggestmns concernmg both con-
tent and language The present art;cle was written under a
grant from the Swiss National Foundation
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mat/che 7 1869), 224-234
Bottazzin~, U Rtemanns E~nflu6 auf E Bett~ und F Casoratt Archive
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VOLUME 2 NUMBER 3 19 98 25
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Neue nschw ande r E Uber dte Wechselwtrku ngen zw~schen der franzo-
sische n S chule, Rte mann u nd Wererstral3 E~ne Ubers tcht m~t zwe~
Quellenstudten
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5 THE MATHEMATICAL NTELLIGENCER
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