Understand the problem

Let \( \Omega = \{ z = x + i y \ \mathbb{C} : |y| \leq 1 \} \). If \( f(z) = z^2 + 2 \) then draw a sketch of $$ f(\Omega ) = \{ f(z) : z \in \Omega \} $$

Justify your answer.

Source of the problem

I.S.I. (Indian Statistical Institute, B.Stat, B.Math) Entrance. Subjective Problem 3 from 2019

Topic

Complex Numbers, Level Curves

Difficulty Level

5 out 10

Suggested Book

Complex Numbers from A to Z by Titu Andreescu

Start with hints

Do you really need a hint? Try it first!

Understand that the map is from x-y plane to u-v plane. (That is the input of the function is from a two-dimensional place and out put is also in ‘another’ two-dimensional plane.

The domain includes all points (x, y) such that \( -1 \leq y \leq 1 \). Here is a picture of the domain.

Domain of f(z)

We know that $$ f(z) = z^2 + 2 $$

Start with z = x + iy.

Then \( f(z) = (x + i y)^2 + 2 = x^2 – y^2 + 2 + 2xy i \)

Hence in the output space:

\( u = x^2 – y^2 + 2 \)

\( v = 2xy \)

Draw level sets. That is, first find what happens at y = 0.

We know that

\( u = x^2 – y^2 + 2 \)

\( v = 2xy \)

Set y = 0 to get

\( u = x^2 + 2 \)

\( v = 0 \)

Hence this is simply the horizontal ray starting at (2, 0) in the u-v plane.

Level set at y = 0

Level set at y = 0

Finally, find the different level curves by setting y = constant (and then varying this constant between -1 and 1)

Since

\( u = x^2 – y^2 + 2 \)

\( v = 2xy \)

Hence \( \frac {v}{2y} = x \)

Replacing in first equation we have

\( u = \frac{v^2}{4y^2} – y^2 + 2 \)

(Notice y is not 0 as we have handled that case previously).

This is a (family of) parabola(s) in the u-v plane with vertex at \( (2 – y^2, 0) \) and opening to the right.

isi entrance 2019 subjective problem 3 solution

 

Play with Geogebra

Connected Program at Cheenta

I.S.I. & C.M.I. Entrance Program

Indian Statistical Institute and Chennai Mathematical Institute offer challenging bachelor’s program for gifted students. These courses are: B.Stat and B.Math program in I.S.I., B.Sc. Math in C.M.I.

The entrances to these programs are far more challenging than usual engineering entrances. Cheenta offers an intense, problem-driven program for these two entrances.

Similar Problems

An inequality with many unknowns

Understand the problemLet be positive real numbers such that . Prove thatSingapore Team Selection Test 2008InequalitiesMediumInequalities by BJ VenkatachalaStart with hintsDo you really need a hint? Try it first!Use the method of contradiction.Suppose that $latex...

Looks can be deceiving

Understand the problemFind all non-zero real numbers which satisfy the system of equations:Indian National Mathematical Olympiad 2010AlgebraMediumAn Excursion in MathematicsStart with hintsDo you really need a hint? Try it first!When a polynomial equation looks...

IMO, 2019 Problem 1 – Cauchyish Functional Equation

This problem is a patient and intricate and simple application of Functional Equation with beautiful equations to be played aroun with.

A sequence of natural numbers and a recurrence relation

Understand the problemDefine a sequence by , andfor For every and prove that divides. Suppose divides for some natural numbers and . Prove that divides Indian National Mathematical Olympiad 2010 Number Theory Medium Problem Solving Strategies by Arthur Engel...

Linear recurrences

Linear difference equationsA linear difference equation is a recurrence relation of the form $latex y_{t+n}=a_1y_{t+n-1}+a_2y_{t+n-2}+\cdots +a_ny_t+b$. If $latex b=0$, then it is called homogeneous. In this article, we shall also assume $latex t=0$ for...

2013 AMC 10B – Problem 5 Maximizing the Difference:

This is based on simple ineqaulities on real numbers.

An inductive inequality

Understand the problemGiven and for all , show that Singapore Mathematical Olympiad 2010 Inequalities Easy Inequalities by BJ Venkatachala Start with hintsDo you really need a hint? Try it first!Use induction. Given the inequality for $latex n=k$, the inequality...

A search for perfect squares

Understand the problemDetermine all pairs of positive integers for which is a perfect square.Indian National Mathematical Olympiad 1992 Number Theory Easy An Excursion in Mathematics Start with hintsDo you really need a hint? Try it first!First consider $latex n=0$....

INMO 1996 Problem 1

Understand the problema) Given any positive integer , show that there exist distint positive integers and such that divides for ; b) If for some positive integers and , divides for all positive integers , prove that .Indian National Mathematical Olympiad...

Trigonometric substitution

Understand the problemLet with . Prove thatDetermine when equality holds.Singapore Team Selection Test 2004InequalitiesMediumInequalities by BJ VenkatachalaStart with hintsDo you really need a hint? Try it first!Show that there exists a triangle $latex \Delta ABC$...