By J Whitesitt
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4). SERIES REPRESENTATION OF ELEMENTARY FUNCTIONS Geometric series Exponential function Trigonometric functions 1 X 1 ¼ zn ¼ 1 þ z þ z2 þ Á Á Á , jzj , 1 1 À z n¼0 The inequality is needed to avoid the singularity (division by 0) at z ¼ 1. 1 n X z z2 ez ¼ ¼ 1 þ z þ þ ÁÁÁ n! 2! n¼0 cos z ¼ 1 X ( À 1)n n¼0 sin z ¼ Hyperbolic functions 1 X n¼0 1 X cosh z ¼ sinh z ¼ Logarithm ( À 1)n z2n z2 z4 ¼ 1 À þ À þÁÁÁ (2n)! 2! 4! z2nþ1 z3 z5 ¼ z À þ À þÁÁÁ (2n þ 1)! 3! 5! z2n z2 z4 ¼ 1 þ þ þ ÁÁÁ ð2nÞ! 2! 4! n¼0 1 X z2nþ1 z3 z5 ¼ z þ þ þ ÁÁÁ ð2n þ 1Þ!
2) y1 ¼ r( cos f cos u þ sin f sin u) ¼ (r cos f) cos u þ (r sin f) sin u y2 ¼ r( sin f cos u À cos f sin uÞ ¼ (r sin f) cos u À (r cos f) sin u (4:1:4) Substituting Eq. 1) into Eq. 4) gives a system of linear equations relating the two coordinate systems in terms of the angle of rotation u; thus y1 ¼ x1 cos u þ x2 sin u y2 ¼ x2 cos u À x1 sin u (4:1:5) The preceding set of equations represents a transformation from the (x1, x2) coordinate system to the (y1, y2) coordinate system. The set of equations in Eq.
13 33 Circular cylindrical coordinate system. 14. The distance from the origin to point P is the spherical radius rs . The azimuth angle f is measured in the x1 – x2 plane from the positive x1 axis. The polar angle us is measured from the positive x3 axis. 14 Spherical polar coordinate system. 4 0 0 rs , 1 us p 0 f 2p EULER’S EQUATION AND HYPERBOLIC FUNCTIONS A fundamental relation between trigonometric and exponential functions is Euler’s equation, which has the form eia ¼ cos a þ i sin a, i¼ pﬃﬃﬃﬃﬃﬃﬃ À1 (2:4:1) Euler’s equation can be derived by comparing the series expansions of each function in Eq.
Principles of modern algebra by J Whitesitt