The ideas of Du Bois-Reymond’s Infinitärcalcül are of great and growing importance in all branches of the theory of functions. With the particular system of notation that he invented, it is, no doubt, quite possible to dispense; but it can hardly be denied that the notation is exceedingly useful, being clear, concise, and expressive in a very high degree. In any case Du Bois-Reymond was a mathematician of such power and originality that it would be a great pity if so much of his best work were allowed to be forgotten..
There is, in Du Bois-Reymond’s original memoirs, a good deal that would not be accepted as conclusive by modern analysts. He is also at times exceedingly obscure; his work would beyond doubt have attracted much more attention had it not been for the somewhat repugnant garb in which he was unfortunately wont to clothe his most valuable ideas. I have therefore attempted, in the following pages, to bring the Infinitärcalcül up to date, stating explicitly and proving carefully a number of general theorems the truth of which Du Bois-Reymond seems to have tacitly assumed—I may instance in particular the theorem of hi. § 2.
2. Scales of infinity in general
3. Logarithmico-exponential scales
4. Special problems connected with logarithmico-exponential scales
5. Functions which do not conform to any logarithmico-exponential scale
6. Differentiation and integration
7. Some developments of Du Bois-Reymond’s Infinitärcalcül
I. General Bibliography
II. A sketch of some applications, with references
III. Some numerical results
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