Complex Numbers and Complex Functions - One Line Questions
1.
What is the product of (a + bi) and (c + di)? —
(ac - bd) + (ad + bc)i
2.
What is the result of adding two complex numbers (a + bi) and (c + di)? —
(a+c) + (b+d)i
3.
The function f(z) = z^2 is analytic everywhere. What are its Cauchy-Riemann equations? —
2x = -2y and 2y = 2x
4.
What is the complex conjugate of a + bi? —
a - bi
5.
What is the Liouville's Theorem in complex analysis? —
A bounded entire function must be constant
6.
What is a branch cut for a multi-valued function? —
A curve or line segment used to define a single-valued branch of the function
7.
The region of convergence for a power series centered at z0 is typically: —
A disk
8.
What is conformal mapping? —
A mapping that preserves angles between curves
9.
What is the branch point of a multi-valued complex function? —
A point around which the function's values do not return to their original value after a circuit
10.
What is a singularity of a complex function? —
A point where the function is not analytic
11.
What is a power series in the complex plane? —
A series of the form sum(a_n * (z-z0)^n)
12.
An entire function is a function that is: —
Analytic everywhere in the complex plane
13.
The part of the Laurent series with negative powers of (z-z0) is called the: —
Principal part
14.
What is the principal value of the argument of a complex number z = a + bi? —
The unique angle theta such that z = |z|(cos(theta) + i sin(theta)) and -pi < theta <= pi
15.
What are the Cauchy-Riemann equations? —
Conditions for a complex function to be analytic
16.
Euler's formula states that e^(ix) is equal to: —
cos(x) + i sin(x)
17.
What is the Laplacian operator, often used in relation to harmonic functions? —
del^2 = d^2/dx^2 + d^2/dy^2
18.
If f(z) = u(x, y) + iv(x, y) is a complex function, what are the Cauchy-Riemann equations in Cartesian coordinates? —
du/dx = dv/dy and du/dy = -dv/dx
19.
Cauchy's Integral Theorem states that if a function f(z) is analytic inside and on a simple closed contour C, then the integral of f(z) around C is: —
Zero
20.
What is the fundamental theorem of algebra in the context of complex numbers? —
Every polynomial of degree n has exactly n roots in the complex numbers (counting multiplicity)
21.
Cauchy's Integral Formula relates the value of an analytic function at a point inside a contour to the integral of the function on the contour. The formula is: —
f(a) = (1 / 2*pi*i) * integral(f(z) / (z-a)) dz
22.
A function that is analytic in a region is also called: —
Holomorphic
23.
What is the Maximum Modulus Principle? —
If f(z) is analytic and non-constant in a bounded domain D, then |f(z)| attains its maximum value on the boundary of D
24.
A complex number is generally represented in the form a + bi, where 'a' and 'b' are real numbers. What is 'a' called? —
Real part
25.
A function f(z) where z is a complex variable is called a complex function. What is the condition for a complex function to be analytic? —
Its derivative must exist at every point in a region
26.
What is the integral of a complex function f(z) along a curve C called? —
Contour integral
27.
Which complex function is often used to map the unit disk to itself conformally? —
Möbius transformation (or fractional linear transformation)
28.
What is the geometric representation of complex numbers called? —
Argand diagram
29.
According to De Moivre's theorem, [r(cos(theta) + i sin(theta))]^n is equal to: —
r^n(cos(n*theta) + i sin(n*theta))
30.
What is the polar form of a complex number z = a + bi? —
r(cos(theta) + i sin(theta))
31.
In the complex number a + bi, what is 'b' called? —
Imaginary part
32.
The modulus of a complex number z = a + bi is denoted by |z| and is calculated as: —
sqrt(a^2 + b^2)
33.
A Laurent series expansion of a function f(z) around an isolated singularity z0 is of the form: —
sum(a_n * (z-z0)^n) + sum(b_n * (z-z0)^-n)
34.
What is the argument of a complex number z = a + bi? —
The angle it makes with the positive real axis in the complex plane
35.
In the polar form r(cos(theta) + i sin(theta)), 'r' represents: —
The modulus of the complex number
36.
What is the residue of a complex function f(z) at an isolated singularity z0? —
The coefficient of the (z-z0)^-1 term in the Laurent series expansion of f(z) around z0
37.
What is the domain of the function f(z) = 1/z? —
The complex plane excluding z=0
38.
A pole is a type of singularity where: —
The function approaches infinity as z approaches the pole
39.
A removable singularity is a point z0 where: —
The function can be defined or redefined at z0 to make it analytic there
40.
An essential singularity is a singularity where: —
The function's behavior near the singularity is very complex (e.g., Casorati-Weierstrass theorem)
41.
In the polar form r(cos(theta) + i sin(theta)), 'theta' represents: —
The argument of the complex number
42.
What is the radius of convergence of a power series? —
The radius of the disk of convergence
43.
What is the imaginary unit, denoted by 'i'? —
The square root of -1
44.
De Moivre's theorem is used to compute: —
The powers and roots of complex numbers
45.
The Residue Theorem provides a powerful way to evaluate contour integrals. It states that the integral of f(z) around a simple closed contour C is equal to: —
2*pi*i times the sum of the residues of f(z) at all singularities inside C
46.
The Schwarz-Christoffel transformation is used to map: —
The upper half-plane onto a polygon
47.
Which of the following is a fundamental property of conformal mappings? —
They are locally equivalent to multiplication by a complex number
48.
What property do the real and imaginary parts of an analytic function satisfy? —
They are harmonic functions
49.
If f(z) = u + iv is analytic, then u and v are harmonic conjugates if: —
u and v satisfy the Cauchy-Riemann equations