TEAM PROJECT. Inverse Trigonometric and Hyperbolic Functions. By definition. the inverse

Chapter 13, Problem 13.1.186

(choose chapter or problem)

Inverse Trigonometric and Hyperbolic Functions. By definition, the inverse sine w = arcsin z is the relation such that sin w = z. The inverse cosine w = arccos z is the relation such that cos w = z. The inverse tangent, inverse cotangent, inverse hyperbolic sine, etc.. are defined and denoted in a similar fashion. (Note that all these relations are multivalued.) Using \(\sin w=\left(e^{i w}-e^{-i w}\right) /(2 i)\) and similar representations of cos w, etc.. show that

(a) \(\arccos z=-i \ln \left(z+\sqrt{\left.z^{2}-1\right)}\right.\)

(b) \(\arcsin z=-i \ln \left(i=+\sqrt{1-z^{2}}\right)\)

(c) \(\operatorname{arccosh} z=\ln \left(z+\sqrt{z^{2}-1}\right)\)

(d) \(\operatorname{arcsinh} z=\ln \left(z+\sqrt{z^{2}+1}\right)\)

(e) \(\arctan z=\frac{i}{2} \ln \frac{i+z}{i-z}\)

(f) \(\operatorname{arctanh} z=\frac{1}{2} \ln \frac{1+z}{1-z}\)

(g) Show that w = arcsin z is infinitely many-valued. and if \(w_{1}\) is one of these values, the others are of the form \(w_{1} \pm 2 n \pi\) and \(\pi-w_{1} \pm 2 n \pi, n=0,1, \cdots\).

(The principal value of w = u + iv = arcsin z is defined to be the value for which \(-\pi / 2 \leqq u \leqq \pi / 2\) if \(v \geqq 0\) and \(-\pi / 2<u<\pi / 2\) if v < 0.)

Text Transcription:

sin w = (e^{iw} - e^{-iw}) /(2i)

arccos z = -i ln (z + sqrt{z^2 -1})

arcsin z = -i ln (i = + sqrt{1 - z^2})

arccosh z = ln (z + sqrt{z^2 - 1})

arcsinh z= ln (z + sqrt{z^2 + 1})

arctan z = i / 2 ln i + z / i - z

arctanh z = 1 / 2 ln 1 + z / 1 - z

w_1

w_1 pm 2 n pi

pi-w_1 pm 2 n pi, n = 0, 1, cdots

-pi/2 leqq u leqq pi/2

v geqq 0

-pi/2 < u < pi/2