English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

PEANO in Mathesis, vol. 9 (1889), p. 75 and p. 110 seems to have been the first to point out that the identical vanishing of the Wronskian of n functions of a single variable is not in all cases a sufficient condition for the linear dependence of these functions.t At the same time he indicated a case in which it is a sufficient condition, j and suggested the importance of finding other cases of the same sort. Without at first knowing of PEANO'S work, I was recently led to this same question, and found a case not included in PEANO'S in which the identical vanishing of the Wronskian is a sufficient condition.? It is my purpose in the present paper to consider these cases and others of a similar nature. By far the most important case in which the identical vanishing of the Wronskian is a sufficient condition for linear dependence is that in which the ftunietions in question are at every point of a certain region analytic functions, whether of a real or complex variable is, of course, immaterial. This case requires no further treatment here. We shall therefore be concerned exclusively with the case in which the independent variable x is real. This variable we will suppose to be confined to an interval Iwhich may be finite or infinite, and if limited in one or both directions muay or may not conltain the end points. In some of the proofs we shall use a subinterval a x_ c b of I; 11 this subinterval we call I' T Whether the functions are real or complex is immaterial. We use the symbol --to denote an identity, i. e., an equality which holds at every point of the interval we are considering.

Certain cases in which the vanishing of the Wronskian is a sufficient condition for linear dependence