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algebraic amplitude angle angular frequency applied mathematics arbitrary constants axis Bessel function boundary conditions calculus Chap column complex numbers compute const convergent series coordinates cosh curve defined denotes derivative determine displacement divergent electromotive force elements equal equation of motion evaluated example expression finite force formula Fourier series fundamental G. N. Watson geometric series given harmonic Hence imaginary indeterminate forms infinite initial conditions inverse transform Laplace transform Laplace's equation Laplacian Laplacian transform Let us consider line integral Maclaurin series mass matrix maximum method multiply natural frequencies obtain operator oscillations particle plane polynomial potential power series problem quantities roots satisfies scalar series converges series expansion shown in Fig sinh solution solve Substituting surface table of transforms temperature theorem theory uniform convergence unit length values vanish vector velocity vibrations write zero