Complex Variables and Applications (8E) - Brown/Churchill. Ex 31.6, 7, 8, 9: Branches/Derivatives

Описание: Complex Variables and Applications (8th Ed) - James Ward Brown and Ruel V. Churchill
Ch 3: Elementary Functions
30: The Logarithmic Function
31: Branches and Derivatives of Logarithms

Remark: Note that different textbooks adopt different notations. In this textbook, arg z is the argument of the complex number z and Arg z is the principal value of the argument of z defined in (-pi, pi]. And log z is the multiple-valued logarithmic function and Log z is the principal branch of log z defined in (-pi, pi]. The relations
arg z = Arg z + 2n pi,
log z = Log z + i(2n pi),
where n ranges over all the integers.

Ex 6: Given that the branch log z = ln r + i theta (r greater than 0, theta in (alpha, alpha + 2pi) of the logarithmic function is analytic at each point z in the stated domain, obtain its derivative by differentiating each side of the identity
e^{log z} = z, (z not 0)
and using the chain rule.

Ex 7: Find all roots of the equation log z = i pi/2.

Ex 8: Suppose that the point z = x + iy lies in the horizontal stipe y in (alpha, alpha + 2pi). Show that when the branch log z = ln r + i theta (r greater than 0, theta in (alpha, alpha + 2pi) of the logarithmic function is used, log(e^z) = z.

Ex 9: Show that
(a) the function f(z) = Log(z - i) is analytic everywhere except on the portion x not greater than 0 of the line y = 1;
(b) the function
f(z) = Log(z + 4) /(z^2 + i)
is analytic everywhere except at the points (1 - i)/sqrt{2}, -(1 - i)/sqrt{2} and on the portion x not greater than - 4 of the real axis.