Algebra II. Ring Theory: Ring Theory by Carl Faith

By Carl Faith

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As () takes values in the intervals 2nl3 ~ () ~ nand 4nl3 ~ () ~ 5n13, two further identical loops are produced symmetrical about the half-lines () = 5nl6 and () = 3nl2 respectively. The curve, a rose-curve called a rhodonea, is shown in Fig. 3(c). The directions of the tangents to the curve at the pole 0 are given by sin 3() = O. The tangents are the half-lines () = 0, () = n13, () = 2n13, () = n, () = 4nl3 and () = 5n13. The points on the curve at which tangents parallel to the initial line occur are determined by finding the values of () for which r sin (), that is 40 sin 30 sin 0, has stationary values.

The graph of the functions y = ex 2 and y = e- x 2 are shown in Figs. 4(c) and (d), respectively. (a) eX (b) x x (e) (d) (0,1) (0 ,1) o x x Fig. 5 Hyperbolic functions There are certain combinations of exponential functions which have properties with a close analogy to those of the circular functions. They are called the hyperbolic sine, the hyperbolic cosine, etc. ---cosh x eX + e X 1 + e 2x e2x + I' 34 Curve sketching We note that cosh x and sech x are even functions and sinh x , cosech x , tanh x and coth x are odd functions .

X, cosh"! x , tanh"! x, etc. respectively. Note that the graphs in Fig. 5(b) are obtained by reflecting those in Fig. 5(a) in the line y = x. The graphs show that COSH- 1 x is two-valued, but that sinh" x and tanh" x are functions of x . (a) v (b) v Fig. 5 Working exercise Sketch the graphs of y = cosech x, y = sech x and y = coth x. Use your graphs to sketch the graphs of the inverse relations y = cosech" x , y = SECH- 1 x and y = coth " x . Note : The curve y = c cosh(x/c) is called a catenary and is the form assumed by a uniform flexible chain suspended between two points and hanging in a vertical plane under gravity.

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