Complex Numbers Overview

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Questions and Answers

What is the integral of f(z) = -i/z?

ln(z) + c
-ln(z) + c
i ln(z) + c
-i ln(z) + c (correct)

A conformal mapping preserves angles between curves.

True (A)

What are the points called where the derivative of an analytic function f'(z) = 0?

Critical points

In a linear fractional transformation, the points that map onto themselves are called ______ points.

invariant Signup and view all the answers

Match the terms with their correct definitions:

Conformal mapping = A mapping that preserves angles between curves Linear fractional transformation = A transformation of the form w = (az + b) / (cz + d), where a, b, c, d are constants and ad - bc ≠ 0 Invariant point = A point that maps onto itself under a transformation Critical point = A point where the derivative of an analytic function is 0 Signup and view all the answers

Which of the following expressions represents a circle centered at the origin with radius 1 unit?

|z|^2 = 1 (A), |z| = 1 (C) Signup and view all the answers

The expression |z - z0| < r represents an open circular disc.

True (A) Signup and view all the answers

What does |z - z0| = 0 represent in terms of points on the complex plane?

It represents a single point at (x0, y0) Signup and view all the answers

The region defined by |z - z0|^2 < r^2 is called an ______ region.

annulus Signup and view all the answers

Match the following complex number expressions with their corresponding geometric representations:

|z - 2| = 3 = Circle centered at 2 with radius 3 Im(z) = 4 = Horizontal line with y-coordinate 4 Re(z) > 0 = Right half plane |z - 1 - i| < 2 = Open disk centered at 1 + i with radius 2 Signup and view all the answers

Which of the following is the correct form of a complex function f(z) in terms of its real and imaginary parts?

f(z) = u(x, y) + iv(x, y) (C) Signup and view all the answers

A function f(z) is said to be ______ at a point z0 if f(z0) is defined and lim(z->z0) f(z) = f(z0).

continuous Signup and view all the answers

The derivative of a complex function f(z) is given by f'(z0) = lim(Δz->0) [f(z0 + 2Δz) - f(z0) ] / Δz.

False (B) Signup and view all the answers

Match the following terms with their definitions:

Analytic function = A function differentiable at all points in its domain. Cauchy-Riemann equations = Conditions that must be satisfied for a function to be analytic. Non-analytic function = A function that is not differentiable at some points in its domain. Signup and view all the answers

What are the two Cauchy-Riemann equations?

ux = vy, uy = -vx Signup and view all the answers

The function f(z) = 1/z is analytic for all complex numbers except z = 0.

False (B) Signup and view all the answers

What is the general form of a complex power series for an analytic function f(z)?

f(z) = co + c1z + c2z^2 + ... + cnz^n Signup and view all the answers

Which of the following functions is NOT analytic?

f(z) = 1/z (C) Signup and view all the answers

Which of the following is NOT a necessary condition for a function to be analytic?

The function must be defined and continuous only in a small neighborhood of the point z. (A) Signup and view all the answers

The function f(z) = |z|^2 is analytic.

False (B) Signup and view all the answers

What are the Cauchy-Riemann equations in Cartesian form?

ux = vy and uy = -vx Signup and view all the answers

In polar form, the Cauchy-Riemann equations are given by ur = and vθ = .

1/r * vθ Signup and view all the answers

Match the following functions with their respective analyticity status:

f(z) = z^3 = Analytic f(z) = |z|^2 = Not analytic f(z) = ln|z| + i arg z = Analytic f(z) = z^2 = Analytic Signup and view all the answers

The function f(z) = ln|z| + i arg z is analytic for all values of z.

False (B) Signup and view all the answers

Explain how the concept of limits is used to derive the Cauchy-Riemann equations.

The Cauchy-Riemann equations are derived by considering the limit of the difference quotient of a complex function as Δz approaches zero along different paths in the complex plane. By choosing two specific paths, one parallel to the real axis and the other parallel to the imaginary axis, and equating the resulting limits, we obtain the Cauchy-Riemann equations. Signup and view all the answers

The image of |z| ≤ 1 in the w-plane is calculated by finding the image of a ______ in the z-plane.

circle Signup and view all the answers

The invariant point in the w-plane is found by solving the equation w = (2 – 2i(z + 1)) / (z + 1) for z.

True (A) Signup and view all the answers

What is the quadratic formula used to solve for the roots of the equation z^2 + (1 + 2i)z + (2i - 2) = 0?

z = (-b ± √(b^2 - 4ac)) / 2a Signup and view all the answers

Match the following terms with their corresponding descriptions:

Invariant point = A point that remains unchanged under a transformation Transformation = A function that maps points from one space to another Quadratic Formula = A formula used to solve for the roots of a quadratic equation Image = The result of a transformation applied to a point or set of points Signup and view all the answers

Which of the following is the correct transformation that maps (0, 1, i) onto (1, 0, 2i)?

w = 1/2 * (z^3 – z^2 – 2iz – 2z + 2i) / (z^2 - 4) (A) Signup and view all the answers

What are the invariant points found using the quadratic formula, z = (-(1 + 2i) ± √((1 + 2i)^2 - 4(1)(2i - 2))) / (2(1))?

z = -i, z = -1 - 2i (C) Signup and view all the answers

How is the denominator of the transformation equation simplified?

The denominator is simplified by using the identity i^2 = -1. Signup and view all the answers

The transformation w = 1/2 * (z^3 – z^2 – 2iz – 2z + 2i) / (z^2 - 4) maps the points (0, 1, i) onto (1, 0, 2i) and therefore verifies the theorem.

True (A) Signup and view all the answers

What are the invariant points of the transformation `w = (z + 1) / (z - 1)`?

The invariant points are <code>1 + √2</code> and <code>1 - √2</code>. Signup and view all the answers

The equation `z^2 - 2z - 1 = 0` represents a quadratic equation that is used to find the ______ points of the transformation `w = (z + 1) / (z - 1)`.

invariant Signup and view all the answers

What is the correct simplification of the expression `(w - 1)(0 + i) / (w + i)(0 - 1)` after dividing the numerator and denominator by `(i + 1)`?

(w - 1)(i - 1) / (w + i)(-i - 1) (C) Signup and view all the answers

The transformation `w = (z + 1) / (z - 1)` maps the points `z1 = 1, z2 = i, z3 = -1` onto the points `w1 = 1, w2 = 0, w3 = -i`.

True (A) Signup and view all the answers

What is the general formula for the LFT/BLT that maps three distinct points `z1, z2, z3` onto three distinct points `w1, w2, w3` ?

The formula is: <code>(w – w1)(w2 – w3) / (w – w3)(w2 – w1) = (z – z1)(z2 – z3) / (z – z3)(z2 – z1)</code> Signup and view all the answers

Match the following transformations to their corresponding invariant points.

<code>w = (z + 1) / (z - 1)</code> = 1 + √2, 1 - √2 <code>w =(2z - 2 + iz - i)/ 2 = (2 + i)z - (2 + i)/2</code> = No invariant point <code>w = z + 2</code> = No invariant point Signup and view all the answers

Explain why the transformation `w = z + 2` does not have any invariant points.

There are no invariant points because the transformation is a translation, shifting all complex numbers by 2 units to the right. No point remains in its original position. Signup and view all the answers

The image of `|z|≤1` under the transformation `w = (z + 1) / (z - 1)` is the ______ of the complex plane.

exterior Signup and view all the answers

Flashcards

Magnitude of z

|z| is the distance from the point z to the origin.

Argument of z

θ (argz) is the directed angle from positive x axis to OP.