Round 7: Tossup 2

Large solutions to these functions are described by a McMahon-type (“muck-MAN type”) asymptotic expansion. Evaluating these functions at “x times e to the 3 pi i over 4” and then taking the real and imaginary parts yields a set of functions denoted “ber,” “bei,” “ker,” and “kei,” which are collectively named for Kelvin. Given an order n, these mutually orthogonal functions have Taylor coefficients with the denominator “gamma of k-plus-one times gamma of k plus n plus one.” (10[1])These functions were introduced (10[1])by Daniel Bernoulli to model oscillations of a heavy chain. When (-5[1])these functions have (-5[1])half-integer order, (-5[1])they arise as spherical (10[1])solutions to the Helmholtz equation. For 10 points, (-5[1])name these functions that are solutions to (10[1])Laplace’s (-5[1])equation (10[2]-5[1])in cylindrical (-5[1])coordinates (10[1])and are symbolized “J-sub-n” (10[2])and (10[1])“Y-sub-n” for their “first” and (10[1])“second” (10[1])kinds. ■END■ (10[3]0[7])

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