Analytic functions - limits, continuity, polynomials and rational functions, power series, Maclaurin series, uniform convergence, Abel's limit theorem - One Line Questions
1.
Let f(z) = Σ (z/2)ⁿ. This is a geometric series. For what values of z does it converge? —
|z| < 2
2.
Consider the power series Σ n! zⁿ. What is its radius of convergence? —
0
3.
According to Abel's Limit Theorem, if Σ aₙ converges, then what is the limit of Σ aₙ zⁿ as z approaches 1 from the left (z → 1⁻)? —
Σ aₙ
4.
What is the radius of convergence of the power series Σ (zⁿ / n!)? —
∞
5.
What is the radius of convergence of the Maclaurin series for f(z) = 1/(1-z)? —
1
6.
What is the Maclaurin series for the function f(z) = 1/(1-z)? —
1 + z + z² + z³ + ...
7.
Let f(z) = e^z. This function is analytic. What is its Maclaurin series? —
1 + z + z²/2! + z³/3! + ...
8.
If f(z) = x² - y² + i(2xy), what is f'(z)? —
2z
9.
If f(z) = u(x, y) + iv(x, y) is analytic, and v(x, y) = x² + y², what is u(x, y)? —
-2xy
10.
If f(z) = u(x,y) + iv(x,y) is analytic, and u(x,y) = x² - y², what is v(x,y)? —
2xy
11.
If f(z) is analytic in a simply connected domain D, and γ is a simple closed contour in D, then ∫γ f(z) dz = ? —
0
12.
What is the derivative of f(z) = z³ at z = 1 + i? —
3(1+i)²
13.
What is a power series centered at z₀? —
A series of the form Σ aₙ (z - z₀)ⁿ
14.
What is the Maclaurin series of a function f(z)? —
A Taylor series centered at z = 0.
15.
What is the domain of analyticity for the function f(z) = 1 / (z² + 1)? —
All complex numbers except z = i and z = -i.
16.
The function f(z) = Re(z) is: —
Not analytic anywhere.
17.
Which theorem states that if a function is analytic in a region, its real and imaginary parts satisfy Laplace's equation? —
Mean Value Theorem for Harmonic Functions
18.
The theorem that states if f'(z) exists in a region, then f is analytic in that region is: —
Cauchy-Riemann Theorem
19.
Which theorem relates the uniform convergence of a sequence of functions to the continuity of the limit function? —
Uniform Convergence Theorem
20.
If a power series converges to f(z) in a disk |z - z₀| < R, then f(z) is analytic in that disk. This is a consequence of: —
The properties of uniform convergence
21.
If {fₙ(z)} converges uniformly to f(z) on S, and each fₙ is continuous on S, then f(z) is: —
Continuous
22.
Let f(z) = Σ aₙ (z-z₀)ⁿ be a power series with radius of convergence R > 0. Then f(z) is: —
All of the above.
23.
If a sequence of functions {fₙ(z)} converges uniformly to f(z) on a compact set K, and each fₙ is analytic on K, then f(z) is: —
Analytic on K
24.
If a sequence of analytic functions {fₙ(z)} converges uniformly to f(z) on a region R, then f(z) is: —
Analytic on R
25.
If a sequence of analytic functions {fₙ(z)} converges uniformly to f(z) on a region R, then the sequence of their derivatives {fₙ'(z)}: —
Converges uniformly to some function on R.
26.
If f(z) = u(x, y) + iv(x, y) is analytic, and f'(z) = 0 in a region, what can be said about f(z)? —
f(z) is a non-zero constant.
27.
Uniform convergence of a sequence of functions {fₙ(z)} to f(z) on a set S means: —
For every ε > 0, there exists N such that if n > N, then |fₙ(z) - f(z)| < ε for all z in S.
28.
Let f(z) = u(x,y) + iv(x,y). If f(z) is analytic, then u and v are called: —
Harmonic conjugates
29.
If a power series Σ aₙ (z-z₀)ⁿ has radius of convergence R, what can be said about the series at |z-z₀| = R? —
It may converge or diverge.
30.
Consider the power series Σ zⁿ/n. It converges for |z| < 1. What is its behavior at z=1? —
It diverges.
31.
Consider the power series Σ zⁿ. It converges for |z| < 1. What happens at z = 1? —
It diverges.
32.
Which of the following is a necessary condition for a function to be analytic in a region? —
It must be differentiable at every point in the region.
33.
Consider the function f(z) = z². Is this function analytic? —
Yes, because it is differentiable everywhere.
34.
Consider f(z) = z². Is it analytic at z=0? —
Yes, because its partial derivatives satisfy the Cauchy-Riemann equations.
35.
Which of the following is NOT a polynomial in z? —
P(z) = z + 1/z
36.
If a power series converges for |z - z₀| < R and diverges for |z - z₀| > R, what is R called? —
Radius of convergence
37.
What is the fundamental condition for a complex function f(z) to be analytic at a point z₀? —
The function must be differentiable in a neighborhood of z₀.
38.
What is the radius of convergence of a power series Σ aₙ (z - z₀)ⁿ? —
The largest radius R such that the series converges for |z - z₀| < R.
39.
Abel's Limit Theorem states that if a power series Σ aₙ zⁿ converges at a point z₀ on its circle of convergence, then: —
The function f(z) = Σ aₙ zⁿ is continuous at z₀.
40.
Cauchy's Integral Formula states that f(z₀) = (1/2πi) ∫γ f(z) / (z - z₀) dz, where γ is a simple closed contour enclosing z₀. This implies: —
The value of an analytic function inside a contour is determined by its values on the contour.
41.
If a function f(z) = u(x, y) + iv(x, y) is analytic in a region, what conditions must its real and imaginary parts satisfy? —
They must satisfy the Cauchy-Riemann equations.
42.
Let fₙ(z) = z/n for n = 1, 2, 3, ... . This sequence converges uniformly to f(z) = 0 on any bounded set. —
True, for any bounded set.
43.
Which condition implies that f(z) = u(x, y) + iv(x, y) is analytic? —
u and v are continuously differentiable and satisfy Cauchy-Riemann equations.
44.
If f(z) = x + iy, is f(z) analytic? —
No, because it does not satisfy Cauchy-Riemann equations.
45.
A rational function is defined as the ratio of two polynomials. Is every rational function analytic? —
No, it is not analytic at the roots of the denominator polynomial.
46.
Let f(z) = |z|². Is this function analytic? —
Yes, it is analytic only at z=0.
47.
The Maclaurin series for sin(z) is: —
z - z³/3! + z⁵/5! - ...
48.
Consider the rational function R(z) = 1/z. At which point is it not analytic? —
z = 0
49.
The Maclaurin series for e^z is given by: —
Σ (zⁿ / n!) from n=0 to ∞