Complex Logarithms and Principal Values - One Line Questions
1.
What is the principal value of Log(e^(i3π/2))? —
-iπ/2
2.
What is the principal value of Log(-i)? —
ln(1) - iπ/2
3.
What is the principal value of Log(e^(i(3π/2)))? —
-iπ/2
4.
The condition for the principal value of the argument Arg(z) is that the angle lies in which interval? —
(-π, π]
5.
Calculate the principal value of the complex logarithm of -1. —
ln(1) + iπ
6.
Calculate Log(1). —
0
7.
What is the principal value of Log(z) when z = e^(i(2π))? —
0
8.
What is the principal value of Log(z) when z = e^(i(-2π))? —
0
9.
What is the principal value of Log(-e)? —
ln(e) + iπ
10.
What is the derivative of Log(z) with respect to z? —
1/z
11.
What is the principal value of Log(e^(i(-π + 0.1)))? —
i(-π + 0.1)
12.
What is the principal value of Log(e^(i(π + 0.1)))? —
i(π + 0.1 - 2π)
13.
The general logarithm log(z) can be thought of as a set of values. How many values are there for each non-zero z? —
Infinitely many
14.
The complex number z = 0 has an undefined complex logarithm because: —
Its modulus is 0, and ln(0) is undefined.
15.
What is the principal value of Log(i^2)? —
ln(1) + iπ
16.
Calculate Log(e^(iπ)) —
ln(1) + iπ
17.
What is the principal value of the complex logarithm of i? —
ln(1) + iπ/2
18.
What is the principal value of Log(e^(iπ/2))? —
iπ/2
19.
Calculate the principal value of Log( (1+i) / (1-i) ). —
ln(1) + iπ/2
20.
What is the principal value of Log(e^(i(5π/2)))? —
iπ/2
21.
Which value of k corresponds to the principal value of the complex logarithm? —
k = 0
22.
If z = x + iy, what is the principal value of Log(z) in terms of x and y? —
ln(√(x² + y²)) + i arctan(y/x) (adjusted for quadrant)
23.
If z = -1 - i, what is the principal value of Log(z)? —
ln(√2) - i3π/4
24.
What is the principal value of Log(1 - i)? —
ln(√2) - iπ/4
25.
What is the principal value of the complex logarithm of 1 + i? —
ln(√2) + iπ/4
26.
If z = -√3 - i, what is the principal value of Log(z)? —
ln(2) - i5π/6
27.
If z = -2, what is the principal value of Log(z)? —
ln(2) + iπ
28.
If z = re^(iθ), what is the principal value of log(z)? —
ln(r) + iθ, where -π < θ ≤ π
29.
What is the principal value of Log(z) when z is a positive real number? —
ln(z)
30.
What is the definition of the complex logarithm of a complex number z? —
ln|z| + i arg(z)
31.
What is the principal value of Log(z) when z is a negative real number? —
ln|z| + iπ
32.
Which of the following is equal to Log(z₁/z₂) when z₁ = 1 and z₂ = i? —
Log(1) - Log(i)
33.
What is the principal value of the complex logarithm, denoted as Log(z)? —
Log(z) = ln|z| + i Arg(z)
34.
Which of the following is NOT a property of the general complex logarithm log(z)? —
log(z^n) = n log(z) for any integer n
35.
What is the general form of the complex logarithm of a non-zero complex number z? —
log(z) = ln|z| + i(arg(z) + 2kπ), where k is an integer
36.
What is the relationship between the general logarithm and the principal value of the logarithm? —
log(z) = Log(z) + 2kπi for some integer k
37.
Consider Log(z₁/z₂). Which relation is generally true? —
Log(z₁/z₂) = Log(z₁) - Log(z₂)
38.
Which property of logarithms holds for the general complex logarithm log(z₁z₂)? —
log(z₁z₂) = log(z₁) + log(z₂)
39.
What is the general form of log(z^n) for a non-zero complex number z and integer n? —
n log(z) = ln|z^n| + i(arg(z^n) + 2mπ)
40.
Consider the function f(z) = Log(z). Is it differentiable at z = -1? —
No, because the principal argument is discontinuous at z = -1.
41.
Does the property log(z₁z₂) = log(z₁) + log(z₂) hold for the principal value Log(z)? —
Not always, due to the principal argument restriction.
42.
The complex logarithm is a multi-valued function. What is the reason for this multi-valued nature? —
The argument of the complex number is periodic.
43.
The function f(z) = Log(z) is continuous everywhere except on which ray emanating from the origin? —
The negative real axis
44.
What is the principal value of the argument of a complex number z, denoted as Arg(z)? —
The value of arg(z) such that -π < Arg(z) ≤ π
45.
Consider the identity Log(z₁z₂) = Log(z₁) + Log(z₂). When does this identity fail? —
When Arg(z₁) + Arg(z₂) is outside the interval (-π, π]
46.
What is the value of e^Log(z) for any non-zero complex number z? —
z
47.
For which complex number is the principal value of the complex logarithm undefined? —
z = 0
48.
What is the value of Log(e^z) for any complex number z? —
z, if -π < Im(z) ≤ π
49.
The range of the principal argument Arg(z) is (-π, π]. What is the argument of z = -1? —
π