Limits, Continuity and Differentiability - One Line Questions
1.
Consider f(z) = 1/z. Find its derivative for z ≠ 0. —
-1/z^2
2.
If f(z) = u + iv, and f'(z) exists, then f'(z) can be expressed in terms of partial derivatives as: —
∂u/∂x + i(∂v/∂x)
3.
What are the Cauchy-Riemann equations for a complex function f(z) = u(x, y) + iv(x, y)? —
∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x
4.
For f(z) = Im(z) = y, the Cauchy-Riemann equations are ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x. What are the values for u=y, v=0? —
0 = 0 and 1 = 0
5.
For f(z) = Re(z) = x, the Cauchy-Riemann equations are ∂u/∂x = ∂v/∂y and ∂u/∂y = -∂v/∂x. What are the values for u=x, v=0? —
1 = 0 and 0 = 0
6.
For f(z) = x - iy, calculate the partial derivatives: ∂u/∂x, ∂u/∂y, ∂v/∂x, ∂v/∂y. —
1, 0, 0, -1
7.
What is the derivative of f(z) = ln(z) for z ≠ 0? —
1/z
8.
Consider f(z) = z^2. Find its derivative. —
2z
9.
If f(z) is analytic in a region R, then f'(z) is also: —
Analytic in R.
10.
Which theorem states that if a function is analytic in a region, then its derivatives of all orders exist and are analytic in that region? —
Cauchy's Integral Theorem
11.
If a complex function f(z) is differentiable at a point z0, then it must be: —
Continuous at z0.
12.
If f(z) is continuous at z0 and g(w) is continuous at w0 = f(z0), then the composite function g(f(z)) is: —
Continuous at z0.
13.
A complex function f(z) is called analytic at a point z0 if it is: —
Differentiable at z0 and in some neighborhood around z0.
14.
If the Cauchy-Riemann equations hold at a point z0 and the partial derivatives ∂u/∂x, ∂u/∂y, ∂v/∂x, ∂v/∂y are continuous in a neighborhood of z0, then f(z) is: —
Differentiable at z0.
15.
Consider f(z) = e^z. Find its derivative. —
e^z
16.
If a function f(z) is analytic in a region R, it means it is differentiable at: —
Every point in R.
17.
What is the definition of the derivative of a complex function f(z) at a point z0? —
f'(z0) = lim_{Δz->0} [f(z0 + Δz) - f(z0)] / Δz, provided the limit exists.
18.
What is the definition of a limit of a complex function f(z) as z approaches z0? —
f(z) approaches L if for every epsilon > 0, there exists a delta > 0 such that |f(z) - L| < epsilon whenever |z - z0| < delta.
19.
Which of the following is NOT a necessary condition for a function f(z) to be differentiable at z0? —
f(z) must be analytic in a neighborhood of z0.
20.
If lim_{z->z0} f(z) = L, what can be said about the value of f(z0)? —
f(z0) can be any complex number, or f(z) may be undefined at z0.
21.
If lim_{z->z0} f(z) = L and lim_{z->z0} g(z) = M, what is lim_{z->z0} [f(z) * g(z)]? —
L * M
22.
If lim_{z->z0} f(z) = L and lim_{z->z0} g(z) = M, and M ≠ 0, what is lim_{z->z0} [f(z) / g(z)]? —
L / M
23.
If lim_{z->z0} f(z) = L, and lim_{z->z0} g(z) = M, what is lim_{z->z0} [f(z) + g(z)]? —
L + M
24.
What is the condition for a function to be continuous at a point z0 in the complex plane? —
lim_{z->z0} f(z) = f(z0)
25.
A complex function f(z) is continuous at a point z0 if which of the following conditions are met? —
lim_{z->z0} f(z) exists, f(z0) is defined, and lim_{z->z0} f(z) = f(z0).
26.
If f(z) = x - iy, where z = x + iy. Is f(z) differentiable anywhere? —
No
27.
Do the Cauchy-Riemann equations hold for f(z) = x - iy? —
No, because ∂u/∂x ≠ ∂v/∂y (1 ≠ -1).
28.
Consider the function f(z) = arg(z). Is this function continuous everywhere in its domain? —
No, it has discontinuities along the negative real axis.
29.
Consider f(z) = Re(z) = x. Is this function differentiable? —
No, it is not differentiable anywhere.
30.
Consider f(z) = Im(z) = y. Is this function differentiable? —
No, it is not differentiable anywhere.
31.
If f(z) is continuous on a set S, does it imply that f(z) is uniformly continuous on S? —
Not necessarily; continuity does not imply uniform continuity.
32.
Consider f(z) = z^n, where n is a positive integer. Using the limit definition, what is f'(z)? —
nz^(n-1)
33.
Which property of limits is essential for proving the differentiability of sums, products, and quotients of complex functions? —
The limit must exist and be independent of the path.
34.
If f(z) = u + iv is analytic, then the Cauchy-Riemann equations are satisfied. Which of the following is also true about u and v? —
They are harmonic functions.
35.
For the limit of a complex function f(z) to exist as z approaches z0, the limit must be the same regardless of the path taken by z towards z0. —
True
36.
For the derivative of a complex function f(z) to exist at a point z0, the limit defining the derivative must be independent of the path along which Δz approaches 0. —
True
37.
If a complex function f(z) = u(x, y) + iv(x, y) is differentiable at z0 = x0 + iy0, then the Cauchy-Riemann equations must hold at (x0, y0), provided that the partial derivatives of u and v exist at (x0, y0). —
True
38.
A function u(x, y) is called harmonic if it satisfies Laplace's equation: ∇^2 u = ∂^2u/∂x^2 + ∂^2u/∂y^2 = 0. —
True
39.
If f(z) = u + iv is analytic, then u and v are harmonic conjugates. —
True
40.
The chain rule for complex differentiation states that if f is differentiable at z0 and g is differentiable at f(z0), then the composite function G(z) = g(f(z)) is differentiable at z0 and G'(z0) = g'(f(z0)) * f'(z0). —
True
41.
If f(z) is differentiable at z0, then lim_{Δz->0} [f(z0 + Δz) - f(z0)] / Δz exists. —
True
42.
Let f(z) = u(x, y) + iv(x, y). If f(z) is differentiable at z0, then the partial derivatives of u and v must satisfy the Cauchy-Riemann equations at z0. —
True
43.
The converse of the Cauchy-Riemann equations is not always true. That is, satisfying the Cauchy-Riemann equations at a point does not guarantee differentiability unless the partial derivatives are also continuous. —
True
44.
If f(z) and g(z) are differentiable at z0, then f(z) + g(z) is also differentiable at z0. —
True
45.
A function f(z) is continuous at z0 if and only if lim_{z->z0} f(z) = f(z0). —
True
46.
If f(z) = u(x,y) + iv(x,y) is differentiable at z0, then the limit defining f'(z0) must be the same regardless of the manner in which Δz approaches 0. —
True
47.
If a complex function f(z) is continuous on a closed and bounded region in the complex plane, then it is also: —
Bounded on that region.
48.
Consider the function f(z) = |z|^2. Is this function differentiable at z = 0? —
Yes, f'(0) = 0.
49.
Consider the function f(z) = |z|^2. Is this function differentiable at z = 1 + i? —
No, the limit defining the derivative does not exist.