Karl wilhelm theodor weierstrass transform

Names [ edit ]. Transforms of some important functions [ edit ]. Constant Functions [ edit ]. Polynomials [ edit ]. Exponentials, Sines, and Cosines [ edit ]. Gaussian Functions [ edit ]. General properties [ edit ]. Low-pass filter [ edit ]. The inverse transform [ edit ]. Generalizations [ edit ]. Related transforms [ edit ]. See also [ edit ].

Karl wilhelm theodor weierstrass transform

German mathematician — EnnigerlohProvince of WestphaliaKingdom of Prussia. BerlinKingdom of Prussia, German Empire. Biography [ edit ]. Mathematical contributions [ edit ]. Soundness of calculus [ edit ]. Calculus of variations [ edit ]. Other analytical theorems [ edit ]. See also: List of things named after Karl Weierstrass. Students [ edit ].

Honours and awards [ edit ]. Selected works [ edit ]. See also [ edit ]. References [ edit ]. Random House Webster's Unabridged Dictionary. In Helicon Ed. Abington: Helicon. October But the lack of rigor that he detected in all available works on the subject, as well as the fruitlessness of his own efforts to surmount this deficiency, frustrated him to the degree that he decided not to present this course again.

His position concerning the applications of his research was clarified in his inaugural speech at the Berlin Academy on 9 Julyin which he stated that mathematics occupies an especially high place because only through its aid can a truly satisfying understanding of natural phenomena be obtained. To some degree his outlook approached that of Gauss, who believed that mathematics should be the friend of practice, but never its slave.

These lectures were given only out of a sense of obligation, however—not from any interest in the subject; for Weierstrass considered geometric demonstrations to be in very poor taste. If, as has been alleged, he sometimes permitted himself to clarify a point by using a diagram, it was carefully erased. In addition to lecturing, Weierstrass introduced the first seminar devoted exclusively to mathematics in Germany, a joint undertaking with Kummer at the University of Berlin in Here again he developed many fruitful concepts that were frequently used by his students as subjects for papers.

Auditors or participants in the seminar included Paul Bachmann. Hermann Amandus Schwarz, and Otto Stolz. The philosopher Edmund Husserl —insofar as he was a mathematician—was also a student of Weierstrass. Weierstrass was not without his detractors: Felix Kleinfor instance, remarked that he and Lie had merely fought for their own points of view in the seminars.

Doubts were not permitted to arise, and checking was hardly possible since Weierstrass cited very few other sources and arranged his methodical structure so that he was obliged to refer only to himself. Schwarz 3 October :. It is self-evident that any and all paths must be open to a researcher during the actual course of his investigations; what is at issue here is merely the question of a systematic theoretical foundation.

Although Weierstrass enjoyed considerable authority at Berlin, he occasionally encountered substantial resistance from his colleagues; and such criticism hurt him deeply. Determined to prevent such a catastrophe, he resolved to remain in Berlin after all. The choice of his successor and publication of his works were problems still to be resolved—and his successor would have to be endorsed by Kronecker.

He was satisfied with neither the circulating transcripts of his lectures nor with the textbooks that followed his concepts and that he had, to some degree, authorized; and his major ideas and methodology remained unpublished. Inhaving already edited the works of Steiner and Jacobi, Weierstrass decided to publish his own mathematical lifework, assured of the help of the younger mathematicians of his school.

He lived to see only the first two volumes appear in print According to his wishes, volume IV was given preferential treatment, and it appeared in Volume III was published the following year. Inat the age of fifty-five, Weierstrass met the twenty-year-old Russian Sonya Kovalevskywho had come to Berlin from Heidelberg, where she had taken her first semester under Leo Koenigsberger.

Unable to secure her admission to the university, Weierstrass taught her privately; and his role in both her scientific and personal affairs far transcended the usual teacher-student relationship. Yet their friendship did not remain untroubled. Her links with socialist circles. On the other hand, many of his letters to her were unanswered. At one juncture she remained silent for three years.

He was instrumental in her obtaining an appointment as lecturer in mathematics at Stockholm in and a life professorship in mathematics in The misinterpretation of their relationship and her early death in brought him additional physical suffering. During his last three years he was confined to a wheelchair, immobile and dependent.

He died of pneumonia. It contained the Cauchy integral proposition and the Laurent proposition. It was published only fifty-three years later, however, when it became clear that Weierstrass at the age of twenty-six had already had at his disposal the principles of his theory of functions, to the development of which he subsequently devoted his lifework.

Yet his contribution to reestablishing the theory of analytic functions ultimately served only to achieve his final aim: the erection of a general theory of Abelian integrals all integrals over algebraic functions and the consideration of their converse functions, the Abelian functions. This dilemma played itself out internally, with Weierstrass paying little attention to his studies at all for a time.

Eventually he decided to pursue mathematics, showing an immediate aptitude for the field. This position, like those his karl wilhelm theodor weierstrass transform held, was far below his abilities and he soon tired of it. At the same time, he continued working on his research, publishing some papers on elliptic and complex functions in relatively obscure publications.

Unfortunately, the stress of his unloved teaching job and his research began to take a toll on Weierstrass's health, leaving him with frequent attacks of dizziness and nausea. These spells were to recur frequently for much of the rest of his life. Weierstrass finally gained recognition with the publication of a paper on the theory of Abelian functions in August Crelle's journal in Over the next few years a number of European universities fought to attract Weierstrass to join their faculty.

He finally accepted an offer from the University of Berlin, his original university of choice. As a lecturer, Weierstrass excelled, attracting students from all over the world. Her early death was a severe blow to him. Although Weierstrass published very little during his career, many of his findings were announced in his lectures, which were collected and published during his later years and after his death.

A colleague, who noted Weierstrass's reputation as "the father of modern analysis," summarized his achievements: "Weierstrass devised tests for the convergence of series and contributed to the theory of periodic functions, functions of real variables, elliptic functions, Abelian functions, converging infinite products, and the calculus of variations.

He also advanced the theory of bilinear and quadratic forms. Weierstrass died at the age of 81, spending the last three years of his life confined to a wheelchair.