Complex Variables and Applications (8E) - Brown/Churchill. Ex 29.1, 30.1, 2: Exponential/Logarithmic
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Complex Variables and Applications (8th Ed) - James Ward Brown and Ruel V. Churchill
Ch 3: Elementary Functions
29: The Exponential Function
Ex 1: Show that
(a) exp(2 + 3 pi i) = -e^2, exp(2 - 3 pi i) = -e^2;
(b) exp((2 + pi i)/4) = sqrt{e/2} (1 + i);
(c) exp(z + pi i) = -exp z.
30: The Logarithmic Function
31: Branches and Derivatives of Logarithms
Ex 1: Show that
(a) Log(-ei) = 1 - pi i/2;
(b) Log(1 - i) = (1/2) ln(2) - pi i/4.
Ex 2: Show that
(a) log(e) = 1 + 2n pi i, n in Z;
(b) log(i) = (2n + 1/2)pi i, n in Z;
(c) log(-1 + sqrt{3}i) = ln(2) + 2(n + 1/3)pi i, n in Z.
Remark: 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.
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