Advanced Calculus fi..

Chapter 4 Integral Calculus of Functions of Several Variables 2256. The function x = am(y) is defined as the inverse of the function y = F(x) of (4.24).Uslng the results of Problem 5, show that am(y) is defined and continuous for all y andhas the following properties:a) as y increases, am(y) increases:b) am(y + 2K) = am(y) + n;c) 3 = Ji-zGG.7. The functions sn(y), cn(y), dn(y) are defined in terms of the function of Problem 6 bythe equations:sn(y) = sin [am(y)], cn(y) = cos [am(y)], dn(y) = d m .Prove the following identities:a) sn2(y) + cn2(y) = 1; j_ I rb) $sn(y) = cn(y)dn(y):dC) a;~n(~) = -sn(y)dn(y); 41d) sn(y 4K) = sn(y);el cn(y + 4K) = cn(y);f) dn(y + 2K) = dn(y).The functions sn(y), cn(y), dn(y) are called elliptic functions. It should be emphasizedthat they depend on k, in addition to y.8. The error function y = erf (x) is defined by the equationThis function is of great importance in probability and statistics and is tabulated in thebooks mentioned after (4.24). Establish the following properties:a) erf (x) is defined and continuous for all x;b) erf (-x) = -erf (x);c) -1 < erf(x) < 1 forallx.The definite integral lab f ( x) dx is defined with respect to a function f (x) definedover an interval a 5 x 5 b. The double integralwill be defined with reference to a function f (x, y) defined over a closed region Rof the xy-plane. It is necessary further to assume that R is bounded, that is, that Rcan be enclosed in a circle of sufficiently large radius; otherwise, just as in the casewhen a or b is infinite, the integral is improper.The definition of the double integral parallels that of the definite integral. Onesubdivides the region R by drawing parallels to the x and y axes, as in Fig. 4.3. Oneconsiders only those rectangles that are within R and numbers them from 1 to n,denoting by A, A the area of the ith rectangle and by h the maximum diagonal of