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18-516: Wangerin may refer to: Friedrich Heinrich Albert Wangerin (1844-1933), German mathematician Walther Wangerin (1884–1938), German botanist Walter Wangerin, Jr. (1944-2021), American author Wangerin Organ Company , American pipe organ company The German name of Węgorzyno , Poland Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with

36-413: A change of variable becomes a special case of Heun's equation . A more general form of Lamé's equation is the ellipsoidal equation or ellipsoidal wave equation which can be written (observe we now write Λ {\displaystyle \Lambda } , not A {\displaystyle A} as above) where k {\displaystyle k} is the elliptic modulus of

54-519: Is when B ℘ ( x ) = − κ 2 sn 2 ⁡ x {\displaystyle B\wp (x)=-\kappa ^{2}\operatorname {sn} ^{2}x} , where sn {\displaystyle \operatorname {sn} } is the elliptic sine function , and κ 2 = n ( n + 1 ) k 2 {\displaystyle \kappa ^{2}=n(n+1)k^{2}} for an integer n and k {\displaystyle k}

72-556: The Encyklopädie der mathematischen Wissenschaften . In 1909, he wrote an article on optics ( Optik ältere Theorie ) for the physics volume of the same encyclopaedia. Wangerin also played an important role in the reviewing of mathematical papers. From 1869 to 1921, he was coeditor of Fortschritte der Mathematik . Wangerin was elected to the German Academy of Scientists Leopoldina in 1883. From 1906 to 1921, he served as President of

90-508: The University of Berlin in 1876. He taught mathematics to the first year undergraduates. He left the University of Berlin in 1882 and became ordinary professor at the University of Halle-Wittenberg . The chair of ordinary professor had fallen vacant because of the death of Eduard Heine, the former teacher of Wangerin. Wangerin held professorship at Halle for more than thirty five years. During

108-576: The University of Königsberg . He worked under the supervision of German mathematician Franz Ernst Neumann . He competed his doctorate from Königsberg University on 16 March 1866. His doctorate thesis was De annulis Newtonianis . After he completing his doctorate, Wangerin took the examinations to become a school teacher. From 1866 to 1867, he trained at the Friedrichswerdersche Gymnasium, Berlin. From 1867 to 1876, he taught mathematics at several gymnasiums. Wangerin became professor at

126-442: The elliptic modulus, in which case the solutions extend to meromorphic functions defined on the whole complex plane. For other values of B the solutions have branch points . By changing the independent variable to t {\displaystyle t} with t = sn ⁡ x {\displaystyle t=\operatorname {sn} x} , Lamé's equation can also be rewritten in algebraic form as which after

144-452: The Academy. In 1907, he was awarded an honorary degree from Uppsala University . He received many medals, including the 1922 Cothenius medal from the German Academy of Scientists Leopoldina. Wangerin function In mathematics, a Lamé function , or ellipsoidal harmonic function , is a solution of Lamé's equation , a second-order ordinary differential equation . It was introduced in

162-783: The Jacobian elliptic functions and κ {\displaystyle \kappa } and Ω {\displaystyle \Omega } are constants. For Ω = 0 {\displaystyle \Omega =0} the equation becomes the Lamé equation with Λ = A {\displaystyle \Lambda =A} . For Ω = 0 , k = 0 , κ = 2 h , Λ − 2 h 2 = λ , x = z ± π 2 {\displaystyle \Omega =0,k=0,\kappa =2h,\Lambda -2h^{2}=\lambda ,x=z\pm {\frac {\pi }{2}}}

180-567: The academic year 1910-11, he was rector of the university. He retired in 1919. After the retirement, Wangerin continued to live in Halle . He was active in mathematical research. He died on 25 October 1933 in Halle. Wangerin was known for his research on potential theory , spherical functions and differential geometry . He wrote an important two volume treatise on potential theory and spherical functions. Theorie des Potentials und der Kugelfunktionen I

198-688: The corresponding calculations for Mathieu functions , and oblate spheroidal wave functions and prolate spheroidal wave functions ). With the following boundary conditions (in which K ( k ) {\displaystyle K(k)} is the quarter period given by a complete elliptic integral) as well as (the prime meaning derivative) defining respectively the ellipsoidal wave functions of periods 4 K , 2 K , 2 K , 4 K , {\displaystyle 4K,2K,2K,4K,} and for q 0 = 1 , 3 , 5 , … {\displaystyle q_{0}=1,3,5,\ldots } one obtains Here

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216-524: The eigenvalues Λ {\displaystyle \Lambda } is, with q {\displaystyle q} approximately an odd integer (and to be determined more precisely by boundary conditions – see below), (another (fifth) term not given here has been calculated by Müller, the first three terms have also been obtained by Ince ). Observe terms are alternately even and odd in q {\displaystyle q} and κ {\displaystyle \kappa } (as in

234-844: The equation reduces to the Mathieu equation The Weierstrassian form of Lamé's equation is quite unsuitable for calculation (as Arscott also remarks, p. 191). The most suitable form of the equation is that in Jacobian form, as above. The algebraic and trigonometric forms are also cumbersome to use. Lamé equations arise in quantum mechanics as equations of small fluctuations about classical solutions—called periodic instantons , bounces or bubbles—of Schrödinger equations for various periodic and anharmonic potentials. Asymptotic expansions of periodic ellipsoidal wave functions, and therewith also of Lamé functions, for large values of κ {\displaystyle \kappa } have been obtained by Müller. The asymptotic expansion obtained by him for

252-555: The paper ( Gabriel Lamé   1837 ). Lamé's equation appears in the method of separation of variables applied to the Laplace equation in elliptic coordinates . In some special cases solutions can be expressed in terms of polynomials called Lamé polynomials . Lamé's equation is where A and B are constants, and ℘ {\displaystyle \wp } is the Weierstrass elliptic function . The most important case

270-581: The title Wangerin . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Wangerin&oldid=1038980313 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Friedrich Heinrich Albert Wangerin Wangerin

288-402: The upper sign refers to the solutions Ec {\displaystyle \operatorname {Ec} } and the lower to the solutions Es {\displaystyle \operatorname {Es} } . Finally expanding Λ ( q ) {\displaystyle \Lambda (q)} about q 0 , {\displaystyle q_{0},} one obtains In the limit of

306-480: Was born on 18 November 1844 in Greifenberg Pomerania , Prussia (now Gryfice , Poland ). He studied at the gymnasium at Greifenberg and completed his final examination with an "excellent" grade in 1862. In spring 1862, Wangerin entered the University of Halle-Wittenberg , where he studied Mathematics and Physics . He was taught by mathematicians Eduard Heine and Carl Neumann . In 1864 he moved to

324-499: Was published in 1909 and Theorie des Potentials und der Kugelfunktionen II was published in 1921. He studied Wangerin functions . Wangerin was also known for writing of textbooks, encyclopaedias and his historical writings. In 1904, he wrote Theorie der Kugelfunktionen und der verwandten Funktionen, insbesondere der Laméschen und Besselschen (Theorie spezieller, durch lineare Differentialgleichungen definierter Funktionen) on functions such as Lamé function and Bessel function for

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