Are these notes free?

Yes. Fill in the short form on this page and the PDF downloads right away. There is no fee for the notes.

Are these official university notes?

No. These are original Uncodemy revision notes for Complex Analysis, written in a handwritten style. They are not issued by AKTU or any other university, and they do not replace your textbook or class notes.

Which syllabus do these complex analysis notes follow?

They follow the common B.Tech complex variables syllabus used by most Indian universities, and the complex variables part of GATE engineering mathematics. Your university may place this in a different semester or skip conformal mapping, so check your own syllabus.

Do the notes cover contour integration of real integrals?

Yes. Unit 7 covers three common types: integrals over 0 to 2π, rational integrals over the real line, and integrals that need Jordan's lemma.

What is the fastest way to find an analytic function from u?

Use the Milne-Thomson method given in Unit 2. Find f'(z) = u_x(z, 0) - i u_y(z, 0) and integrate with respect to z.

Can I get handwritten notes for another subject?

Yes. Browse the B.Tech notes hub for other subjects, or use the handwritten notes request page to ask for a subject that is not listed yet.

Complex Analysis Handwritten Notes PDF

Key takeaways

Eight units, from polar form to real integrals by residues.
CR equations in Cartesian and polar form, plus the Milne-Thomson method.
Bilinear transformation, fixed points and cross-ratio with a worked example.
Residue formulas and the three types of real integrals solved by contour integration.
Original Uncodemy revision notes; check your own university syllabus.

About these handwritten notes

These are original Uncodemy revision notes for Complex Analysis, written in a clean handwritten style so they feel like a topper's notebook rather than a textbook. They are short on purpose: each unit gives the key definitions, formulas, a solved example or two and the points that are easy to forget.

The notes follow the common syllabus that most Indian B.Tech universities use for this subject, including universities that follow the AICTE model curriculum. They are not official notes from any university. Unit names, order and depth differ from one university to another, so check your own university syllabus and mark any topic that is missing before you rely on the notes for an exam. The notes keep proofs short and focus on the methods you need for numericals: finding f(z), choosing the right theorem and computing residues.

What's inside the complex analysis notes

The notes have eight units. One line on each:

Unit 1: Complex numbers

Conjugate, modulus, polar and exponential form, nth roots and standard complex functions such as e^z, sin z, cos z and log z.

Unit 2: Analytic functions

Analytic and entire functions, the Cauchy-Riemann equations in Cartesian and polar form, harmonic functions and harmonic conjugates, and the Milne-Thomson method, with a solved example.

Unit 3: Conformal mapping

What makes a map conformal, critical points, standard maps (translation, scaling, rotation, inversion, z²) and the bilinear (Mobius) transformation with fixed points and cross-ratio.

Unit 4: Complex integration

Line integrals, Cauchy's theorem, Cauchy's integral formula and its derivative form, Morera's and Liouville's theorems, with solved examples.

Unit 5: Series

Taylor series and Laurent series in an annulus, with a solved Laurent expansion.

Unit 6: Singularities and residues

Removable singularities, poles of order m and essential singularities, zeros, residue formulas and the residue theorem, with an example.

Unit 7: Real integrals by residues

Type 1 integrals over 0 to 2π using z = e^(iθ), Type 2 rational integrals over the real line, and Jordan's lemma for Type 3 integrals.

Unit 8: Most asked questions (practice)

Ten practice questions covering every unit.

How to revise complex analysis with these notes

Start with CR equations. Most of the subject depends on knowing when a function is analytic. Practise checking u_x = v_y and u_y = -v_x on a few functions until it is quick.
Use Milne-Thomson to find f(z). The notes give the shortcut: f'(z) = u_x(z, 0) - i u_y(z, 0), then integrate. It is faster than finding v first.
Draw the contour every time. Before using Cauchy's formula or the residue theorem, mark the singular points and check which ones lie inside the curve. The notes stress this step.
Make a residue formula card. Write the formulas for a simple pole, a pole of order m and the p(z)/q(z) case on one card and revise it daily.
Practise Laurent series by region. The same function gives a different series in each annulus. Do one function in all its regions.
Do one real integral of each type. Type 1, Type 2 and Type 3 each follow a fixed pattern, so one good example of each goes a long way.

Exam focus: high-weightage topics

According to the tip at the end of the notes, GATE mostly asks CR equations, residue values and contour integrals by the residue theorem. University papers add longer questions on analytic functions, bilinear maps, Laurent series and real integrals.

Topics worth extra time:

Harmonic functions: show that u = e^x cos y is harmonic and find the analytic function f(z).
Theory: derive the CR equations, and prove that an analytic function with constant modulus is constant.
Bilinear maps: find the map that sends three given points to three given points.
Cauchy's integral formula: evaluate a contour integral with two singular points inside the curve.
Laurent series of 1/((z - 1)(z - 2)) in 1 < |z| < 2.
Singularities: classify the singularities of functions such as (1 - cos z)/z² and e^(1/z).
Residues at simple and double poles.
Real integrals such as ∫₀^2π dθ / (2 + cos θ).

Typical questions

The practice set also asks you to state Cauchy's integral formula and Liouville's theorem. These are practice questions, not a prediction of your paper.

Who these notes are for

B.Tech students studying complex variables, usually in Engineering Mathematics 2 or 3, depending on the university.
ECE and EE students, who use complex functions and residues again in signals, control systems and electromagnetics.
GATE aspirants revising the complex variables part of engineering mathematics.

For unit-wise practice questions with answer outlines, use the Engineering Mathematics 2 important questions PDF, which has units on complex differentiation and integration.

Common mistakes to avoid

Assuming CR equations are enough. The partial derivatives must also be continuous for the function to be analytic.
Counting poles outside the contour. Only singular points inside the curve contribute to the residue theorem.
Using the simple pole formula at a double pole. Find the order of the pole first, then choose the formula.
Wrong Laurent region. Write the series that matches the annulus given in the question, not the one that is easiest.
Forgetting dz in Type 1 integrals. With z = e^(iθ), dθ = dz/(iz). Missing the i changes the answer.

Key results to know by heart

Before the exam, make sure you can write these from memory without looking at the notes:

The Cauchy-Riemann equations in Cartesian form (u_x = v_y, u_y = -v_x) and in polar form.
Cauchy's theorem, Cauchy's integral formula and the formula for the nth derivative.
The general bilinear transformation and the cross-ratio property.
The Laurent series form and the meaning of the principal part.
Residue formulas for a simple pole, a pole of order m and f = p(z)/q(z).
The substitution z = e^(iθ) with cos θ = (z + 1/z)/2 and sin θ = (z - 1/z)/2i.

A simple 4-day revision plan

Day 1: Units 1 and 2. Complex numbers, CR equations, harmonic functions and Milne-Thomson.
Day 2: Units 3 and 4. Conformal and bilinear maps, Cauchy's theorem and integral formula.
Day 3: Units 5 to 7. Laurent series, residues and real integrals.
Day 4: Solve the ten practice questions with a timer and revise your residue formula card.

Download Complex Analysis Handwritten Notes PDF