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

ISI MStat PSB 2007 Problem 7 | Conditional Expectation

This is a very beautiful sample problem from ISI MStat PSB 2007 Problem 7. It’s a very simple problem, which very much rely on conditioning and if you don’t take it seriously, you will make thing complicated. Fun to think, go for it !!

ISI MStat Entrance Exam books based on Syllabus

Are you preparing for ISI MStat Entrance Exams? Here is the list of useful books for ISI MStat Entrance Exam based on the syllabus.

ISI MStat PSB 2008 Problem 8 | Bivariate Normal Distribution

This is a very beautiful sample problem from ISI MStat PSB 2008 Problem 8. It’s a very simple problem, based on bivariate normal distribution, which again teaches us that observing the right thing makes a seemingly laborious problem beautiful . Fun to think, go for it !!

ISI MStat PSB 2004 Problem 6 | Minimum Variance Unbiased Estimators

This is a very beautiful sample problem from ISI MStat PSB 2004 Problem 6. It’s a very simple problem, and its simplicity is its beauty . Fun to think, go for it !!

ISI MStat PSB 2004 Problem 1 | Games and Probability

This is a very beautiful sample problem from ISI MStat PSB 2004 Problem 1. Games are best ways to understand the the role of chances in life, solving these kind of problems always indulges me to think and think more on the uncertainties associated with the system. Think it over !!

ISI MStat PSB 2013 Problem 5 | Simple Random Sampling

This is a sample problem from ISI MStat PSB 2013 Problem 5 based on the simple random sampling model, finding the unbiased estimates of the population size.

ISI MStat PSB 2013 Problem 4 | Linear Regression

This is a sample problem from ISI MStat PSB 2013 Problem 4. It is based on the simple linear regression model, finding the estimates, and MSEs.

ISI MStat PSB 2011 Problem 1 | Linear Algebra

This is ISI MStat PSB 2011 Problem 1, based on patterns in matrices and determinants, and using a special kind of determinant decomposition. Try this out!

ISI MStat PSB 2014 Problem 9 | Hypothesis Testing

This is a another beautiful sample problem from ISI MStat PSB 2014 Problem 9. It is based on testing simple hypothesis, but reveals and uses a very cute property of Geometric distribution, which I prefer calling sister to Loss of memory . Give it a try !

ISI MStat PSB 2008 Problem 10 | Hypothesis Testing

This is a really beautiful sample problem from ISI MStat PSB 2008 Problem 10. Its based on testing simple, hypothesis. According to, this problem teaches me how observation, makes life simple. Go for it!