It was at Munich that Hamburger undertook research for his doctorate, advised by Alfred Pringsheim. He submitted his 69-page thesis Über die Integration linearer homogener Differentialgleichungen and, after an oral examination conducted by Alfred Pringsheim and Arnold Sommerfeld, he was awarded the degree. Hamburger then went to Paris to continue his studies at the Collège de France with Jacques Hadamard. However, the political situation in Europe was quickly deteriorating at this time with a diplomatic crisis developing following the assassination of Archduke Franz Ferdinand of Austria on 28 June 1914. Austria-Hungary declared war on Serbia on 28 July, Russia began mobilising against Austria-Hungary on the following day, and Germany declared war on Russia on 1 August. Germany attacked Luxembourg on 2 August and on 3 August declared war on France. At this stage Hamburger was a German national in France with the two countries at war. He escaped quickly back to Germany and enlisted in the army. Still in August he was sent to take part in the fighting in Galicia. His brother had trained as a lawyer but in 1915 he also undertook military service.
The German armies in Galicia were supporting the Austro-Hungarian troops in their battles with Russia. Hamburger fought there with the German troops until December 1916 when, because of health problems, he was brought back to Germany to undertake work in aerodynamics. After being assigned to the Erste Deutsche Automobile-Fachschule near Mainz, then to the Institute for Fluid Mechanics and Aerodynamics and of aeronautics in Darmstadt, he was sent to German Experimental Institute for Aviation in Adlershof near Berlin. Hamburger was delighted to be back in Berlin for he was then able to continue undertaking research at the Friedrich-Wilhelm University of Berlin. He submitted his work on solving Stieltjes' moment problem for his habilitation and received his right to teach at university in April 1919. The work of his Habilitationsschrift appeared in four publications, the results were summarised in Über eine Erweiterung des Stieltjesschen Momentenproblems (1919) and given in full in a 3-part paper with the same title (1920, 1920, 1921). The related paper Beitrage zur Konvergenztheorie der Stieltjesschen Kettenbrüche (1919) was submitted from Adlershof in 1918. Margaret Eleanor Grimshaw writes :-
The work on the moment problem incorporated earlier results of Stieltjes, Perron and others, generalized them and gave new proofs and presentations.In 1920 he published Über die Konvergenz eines mit einer Potenzreihe assoziierten Kettenbruchs in Mathematische Annalen. He adds a note explaining how the work was undertaken earlier:-
Please note that I have undertaken this work in the spring of 1916. Prevented by military obligations, I could not finish the manuscript until March 1917 and then sent it soon after to the editors of the Journal of Pure and Applied Mathematics, who accepted the work. In February 1919, I requested that the manuscript be returned because I had realized that my original proof could be significantly simplified.Hamburger taught at the University of Berlin as a docent from 1919, being promoted to Extraordinary Professor on 1 July 1922. He was made a full professor at the University of Cologne on 11 April 1924. Ernst Fischer had been appointed as professor of mathematics at the University of Cologne in 1920. At this time there was only one chair of mathematics at Cologne but a second chair was created by the Prussian Ministry of Science, Art and Education and it was to this second chair that Hamburger was appointed. On 15 October 1925 Hamburger married Martha Jessen, known as Malla. She was the daughter of Peter Jessen (1858-1926), the director of the library of the Berlin Arts and Crafts Museum.
On 30 January 1933 Hitler came to power and on 7 April 1933 the Civil Service Law provided the means of removing Jewish teachers from the universities, and of course also to remove those of Jewish descent from other roles. All civil servants who were not of Aryan descent (having one grandparent of the Jewish religion made someone non-Aryan) were to be retired. However, there was an exemption clause which, among others, exempted non-Aryans who had served in the 1914-18 War. Hamburger came under this exemption clause and so remained in his position in 1933. Other Jewish mathematicians who had served in World War I and should have been exempted were, nevertheless, dismissed. Hamburger continued in post at Cologne until 14 November 1935 when he was granted leave. On 31 December 1935 he was retired from his professorship and he travelled to Berlin where he lived with his widowed mother. On 12 June 1937, Hamburger and his wife were divorced. Not surprisingly, Hamburger published nothing during the six years he spent in Germany after the Nazis came to power. He had published 29 papers up to 1933 with the 91-page paper Ribaucour transformationen und sphärische Abbildung appearing in 1933. His next published work was 3-part Beweis einer Carathéodoryschen Vermutung I, II, III (1940, 1941, 1941).
On 25 March 1939 he submitted a request to the Cologne University Board to transfer his residence to the United States and to continue to receive payment of his pension. He wrote :-
I want to emigrate, but because I am a Jew, it is only possible if I get permission for this from my superiors. From my retirement pension I have for years supported my mother, who is completely dependent on my support ...The National Socialist Executive Chairman of the Board, Erwin Fassl, put many obstacles in Hamburger's way. He wrote to the government minister:-
With regard to the personality of Professor Hamburger I wish to express my opinion that I do not consider him so reliable that I could endorse his application for continued payment of his pension.Hamburger shortly thereafter changed his request to emigrate to the Netherlands. Fassl's objections did not prevent Hamburger being given permission to move to the Netherlands, initially for a period of two years but subject to being recalled at any time. He had to make six-monthly reports to the pension awarding authority. On 14 August 1939 Hamburger left Germany and, instead of going to the Netherlands, travelled directly to Britain. In January 1940 the Ministry in Cologne asked the Cologne University Administration for Hamburger's address. At this point the Ministry was told that he was in England so they immediately stopped his pension payments. Hamburger's mother, in an attempt to have the payments continue, told the Ministry that Hamburger had been travelling in England at the time that war broke out and had been prevented from returning.
On arriving in England, Hamburger was supported by the "Society for the Protection of Science and Learning" and he took up residence in Croydon where he continued his mathematical research. He explained in a letter the circumstances of his flight to England :-
I left Berlin and Germany August 14th and came straight to England where I have been ever since. By this you will see that I did not carry out my original plan of stopping in Holland first and consequently I have difficulties in getting the pension due to me, paid to my mother who, of course, is depending on receiving it.Two weeks after he had arrived in England, Britain and Germany were at war and this made life much harder for "aliens". He corresponded with Hermann Weyl about obtaining a position in the United States, in particular at Stanford University. However, in 1940 Hamburger, as a German citizen, was placed in an internment camp. He wrote to Hermann Weyl, now in Princeton, on 28 November 1940 :-
The letter of August which you mentioned did not reach me since I was in an Internment Camp from 25th of June to 25th of October. I hadn't too bad a time there. I had a lot of mathematical work to do, having had to lecture to very nice scientists some of whom were very well known.Released from the internment camp, he spent three weeks back in Croydon before moving to 45 Owlstone Road, Cambridge. However, the Society for the Protection of Science and Learning could not continue to support him. He wrote to Hermann Weyl on 28 November 1940 :-
As you know I am living here on a grant kindly awarded by the Society for the Protection of Science and Learning. Actually these funds seem to be rather low and might become exhausted within not too long a space of time. The amount of my grant has been considerably reduced so that now I have to live off £12.10 monthly. And I think it is significant that when calling on the Society some days ago I was told to do everything in my power to hasten my immigration to U.S.A.He wrote again to Weyl on 6 January 1941 :-
Dear Weyl, I have to inform you that since my last letter of November the 28th I have been notified by the Society for the Protection of Science and Learning that my grant which expires on January the 31st cannot be renewed as the funds of the Society a running very low. Since it is extremely unlikely that I shall be able to obtain any paid work I shall be without any means on the above date and therefore I am relying entirely upon your help in these very unfortunate circumstances. I hope you will be able to find some way out for me. Sincerely and gratefully, Yours.Weyl replied with a letter that did not offer Hamburger much hope :-
The Natural Science Department of the Rockefeller Foundation seems to find it beyond their means to come to rescue of all the scholars who had found a refuge in the British Empire, and, as I was recently advised, they now have under consideration a plan which, if approved, will afford a type of local assistance to scientists of eminence in England, and until it is determined whether we shall be able to forward such a plan, requests for assistance to scholars in England to come to this country will be held in abeyance.Let us record what happened to other members of Hamburger's family. His brother became a notary in 1927 but was removed from this office in 1935. In 1938 he was forbidden to practice as a lawyer except as a "consultant" to provide legal assistance for Jews. In 1940 he tried to emigrate but had left it too late. Margarethe Hamburger, the mother of Hans and Georg, died in hospital in April 1941 after falling down stairs in Berlin. From September of that year Georg was forced to wear the Jewish star. In June 1943 he was sent to the Theresienstadt concentration camp where he died in August of the following year.
We return to describe Hans Hamburger's life in England. In January 1941 he was appointed as an Assistant Teacher of Mathematics at the Modern and High School in Luton, Bedfordshire. The head teacher was clearly embarrassed at having such a top mathematician teaching school children. She wrote to him on 2 February 1941 :-
My committee of Governors, Mr Godfrey and myself are all very sensible of the tragedy of the position in which such an eminent mathematician as yourself turns schoolmaster. But having said that we will try and forget it and make your interlude with us as happy as possible.He only taught at the school for a term before returning to Cambridge in April 1941. He was there when he received word of his mother's death. In Cambridge, Hamburger had great moral support from G H Hardy. In the autumn of 1941 he was appointed as a Temporary Lecturer at University College Southampton.
Hamburger was not only undertaking research in mathematics after he reached England but he was also writing the pamphlet How Nazi Germany has mobilised and controlled labour. The review  describes the text:-
This valuable exposure of the reintroduction of slavery into Germany, to be followed by something worse in the rest of occupied Europe, is timely. The main part of the system described was introduced into Germany not in war, but as a preparation for a destructive world war and world domination. Mr Hamburger's pamphlet is well documented. The achievement in getting rid of 5,000,000 unemployed in Germany between 1932 and 1940 is shown in its true perspective. About 1,000,000 persons were annihilated or eliminated. It is clear that the system had to lead to war or further murders and revolution.After serving as a Temporary Lecturer at University College Southampton in 1941-42, Hamburger had his contract extended on 15 June 1942 to cover the following academic year 1942-43. On 21 June 1943 he received a permanent position at Southampton and he continued to teach there until 1947. During these years he collaborated with Margaret Grimshaw writing the textbook Linear Transformations in n-Dimensional Vector Space. An Introduction to the Theory of Hilbert Space. The book was published by Cambridge University Press in 1951.
In 1946, after World War II had ended, the University of Cologne approached Hamburger about taking up the chair at Cologne from which he had been dismissed by the Nazis. He indicated that he would like to return to Cologne but wished first to take up a position of full professor at the University of Ankara. Despite the University of Cologne being unhappy that Hamburger tried to keep the Cologne position open while continuing to work in Ankara, he did return to Cologne and take up his former chair on 1 June 1953. He spent 1954-55 as a visiting professor in the United States at Cornell University in Ithaca. Back in Cologne, he was already seriously ill when he married Vera Schereschevsky in the summer of 1956. He died from tuberculosis two months after his wedding.
Margaret Grimshaw, Hamburger's collaborator and colleague at Southampton writes :-
In carrying out his mathematical research, Hamburger formed the habit of working far into the early morning hours and of sleeping in the afternoon. He claimed that this made possible a more intense concentration. In his relaxed and non-mathematical moments he sought society and enjoyed conversation. He was an enthusiastic dancer, skier and climber; he was knowledgeable about modern art, he enjoyed the theatre and, rather late in life, he came to have a great interest in music. As a colleague he was stimulating though exacting; as a companion he was gay and entertaining.She also gives an overview of Hamburger's mathematical contributions :-
Throughout Hamburger's published work there recurs evidence of the effect of his early interest in differential geometry and mathematical methods. He had easy control over complicated analytical manipulations and long-sustained detailed investigations, and he turned again and again to geometrical interpretations to provide both motivating ideas and illustrative examples. The course of his work on each of the main problems that he tackled follows the same pattern; a series of shorter preparatory papers on different aspects of the problem, followed by a very detailed complete account, usually spread over two or three papers. There are two distinct periods of activity, before and after 1942. The work of the first period contains important contributions in two fields; that of differential geometry and partial differential equations, as well as that of continued fractions and the Stieltjes moment problem with which his name is probably most frequently associated. ... After 1942 his work was confined to the investigation of linear transformations in Hilbert space.
Article by: J J O'Connor and E F Robertson