188 CHAPTER 5 • ELEMENTARY FUNCTIONS- Given an elegant argument that explains why t he following functions a.re harmonic.
(a} h (x, y) =sin x cosh y.
(b} h(x,y) = cosxsinhy.
(c} h (x, y) = sinhx cosy.
(d} h (x, y} = cosh x sin y.16. Establish the following identities.(a} e" = cosz + isin z.
(b} cosz = cosxcoshy-isinxsinhy.
(c} sin (z1 + z2} = sin zl cos z2 + cos z 1 sinz2.
(d} !cos zl^2 = cos^2 x + sinh^2 y.
(e} coshz = coshxcosy+isinhxsiny.
(f} cosh^2 z - sinb^2 z = 1.
(g} cosh (z1 + z2} = cosh z1 cosh z2 + sinh z1 sinh z2.- Find the complex impedance Z if
(a} R = 10, L = 10, C = 0.05, and w = 2.
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(b) R = 15 , L = 10, C = 0.05, and w = 4.1 8. Explain how sin z and the function sin x that you studied in calculus are different.
How are they similar?5.5 Inverse Trigonometric and Hyperbolic Functions
We expressed trigonometric and hyperbolic functions in Section 5.4 in terms of
the exponential function. In t his section we look at their inverses. When we
solve equations such as w = sinz for z, we obtain formulas that involve the
logarithm. Because trigonometric and hyperbolic functions are all periodic, t hey
are many-to-one; hence their inverses are necessarily multivalued. The formulas
for the inverse trigonometric functions are
arcsinz = -ilog [iz + (1-z^2 )!),
arccosz = -ilog (z + i (1 - z^2 )! ] , and
arctanz =~log(:~;).(5-45)