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algebra approximation augmented matrix axis Bessel's functions characteristic equation coefficients column complex number conductor constant containing corresponding cosh cosine cosine series coth Cramer's rule cross product csch curve degree density derivative dimensional analysis dimensional formula double root dx dx elements emix equa equal to zero equation in problem example Express the function factor flux Fourier series frequency function shown given equation gives Graeffe's method homogeneous hyperbolic functions integration interchange method minus multiple roots Newton's formula Newton's method obtained odd function odd permutation order determinant original equation polynomial quantity rank real roots root of fix round diaphragm scalar sech shows Simpson's rule sine sinh solve Stirling's formula string substitute symmetric roots synthetic division Table tanh term tion Transform unknowns upper bound variable vector velocity