Many facts about spherical harmonics ( such as the addition theorem ) that are proved laboriously using the methods of analysis acquire simpler proofs and deeper significance using the methods of symmetry.
12.
An "'algebraic addition theorem "'is one in which " G " can be taken to be a vector of polynomials, in some set of variables.
13.
With the addition theorems above and for a given " k " with 0 < " k " < 1 they therefore are solutions to the following nonlinear ordinary differential equations:
14.
William Burnside used such triangles, projecting from a point on the hyperbola " xy " = 1 onto the main diagonal, in his article " Note on the addition theorem for hyperbolic functions ".
15.
Having established these two limits, one can use the limit definition of the derivative and the addition theorems to show that cos " x " } } and " sin " x " } }.
16.
Since this area was poorly represented at Berlin, he moved to the University of Halle, where he obtained his doctorate in 1892 under Albert Wangerin with a dissertation on the topic " Ueber eine gewisse Klasse continuirlicher Gruppen und ihren Zusammenhang mit den Additionstheoremen " ( " On a certain class of continuous groups and their relation to addition theorems " ).
