Mathematics in Physics and Engineering describes the analytical and numerical (desk-machine) methods that arise in pure and applied science, including wave equations, Bessel and Legendre functions, and matrices. The manuscript first discusses partial differential equations, as well as the method of separation of variables, three-dimensional wave equation, diffusion or heat flow equation, and wave equation in plane and cylindrical polar coordinates. The text also ponders on Frobenius' and other methods of solution. Discussions focus on hypergeometric equation, Bessel's equation, confluent hypergeometric equation, and change of dependent and independent variables. The publication takes a look at Bessel and Legendre functions and Laplace and other transforms, including orthogonal properties, applications from electromagnetism, spherical harmonics, and application to partial differential equations. The book also examines matrices, analytical methods in classical and wave mechanics, calculus of variations, and complex variable theory and conformal transformations. The book is a dependable reference for mathematicians, engineers, and physicists both at undergraduate and postgraduate levels.