Complex Analysis Course for CSIR NET and GATE: What to Check

A complex analysis course for CSIR NET and GATE should cover eight blocks: the plane and power series, analytic and harmonic functions, contour integration, Liouville's theorem and its relatives, Taylor and Laurent series, singularities and residues, Rouché's theorem with the argument principle, and Möbius maps. The official syllabi name each one.

Not every exam names every block. The table shows which syllabus names what. The last column is a test you can run on any lecture list. CUET PG is the Common University Entrance Test for Postgraduate admission, and SCQP19 is its Mathematics paper code.

Sources: the CSIR NET Mathematical Sciences syllabus, Unit 2, the Complex Analysis section of the GATE 2027 MA syllabus and the CUET PG Mathematics (SCQP19) syllabus.

The CUET PG complex analysis unit ends at Cauchy's integral formula, Morera's theorem, Liouville's theorem, the fundamental theorem of algebra and harmonic functions. For that paper, blocks 1 to 4 are the ones to check. For the rest of the CSIR NET paper, see our guide to the CSIR NET Mathematical Sciences syllabus.

Free lectures are enough for the theory if you can set your own pace and find your own practice. Pay when you want timed tests with solutions, a fixed schedule and someone to ask. We publish this page, and our own course is one of those compared.

Our recommendation:

• Comfortable with real analysis and good at working alone: use NPTEL for the theory and add past papers.
• Months, not a year, before the exam: take a course with tests and solutions, because lectures alone do not show which questions cost you marks.
• Sitting CUET PG only: you need blocks 1 to 4 and no more.

If you also want a textbook, NPTEL's reference list names Priestley's Introduction to Complex Analysis (Indian edition), Ahlfors, and Brown and Churchill. Our guide to CSIR NET maths books explains how to settle on one main book.

Start with our free study content and the PDF Drive. For CSIR NET and GATE together, our 2027 course costs ₹15,999 (or four instalments of ₹3,999.75) with access through 30 June 2027, as of October 2026, and includes doubt support. A ₹999 Demo Plan with 10 days of access lets you try it first.

Choose the contour from where the singularities sit and how the integrand behaves on the arc. Close a large semicircle when the integrand shrinks fast enough, indent around a pole on the real axis, and use a keyhole only when the integrand has a branch cut. Learners ask exactly this: "What are the guidelines for choosing a contour? Does it matter?" and "Would it be some kind of keyhole contour?"

The semicircle is there because the residue theorem works only on closed curves, and the real line is not closed. The arc closes it and is chosen so that its integral tends to zero.

Worked example. To evaluate the integral of 1/(x² + 1) over the whole real line, close the contour with the upper semicircle of radius R > 1. On the arc, |z² + 1| ≥ R² − 1, so the arc integral is at most πR/(R² − 1), which tends to 0. The only pole inside is z = i, with residue 1/(2i). The integral is 2πi × 1/(2i) = π.

For sin x / x, use e^(iz)/z, indent above the pole at 0, and the small arc contributes −iπ. The integral of e^(ix)/x then equals iπ in the principal-value sense, and its imaginary part gives π.

The GATE MA syllabus names "residue theorem and applications for evaluating real integrals". A course that skips a worked set of these fails block 6 in the table.

Run four checks on any course page, each with a clear pass or fail. They take about ten minutes and cost nothing.

Does the lecture list name every block your exam needs?

Pass: every block your exam names in the table above has a lecture or session named for it. Fail: a block you need appears only inside a general "revision" or "practice" lecture, where you cannot tell what is taught.

Can you watch a full lecture first?

Pass: a complete sample lecture or demo class, long enough to judge the pace and the language. Fail: only a trailer. Digital courses are rarely refundable, so the sample is your only trial.

Are there timed tests with worked solutions?

Pass: tests that cover each block your exam names, in the question formats of your exam, each followed by a solution walkthrough. Fail: practice sheets with answers only. For how the papers are built, see our guide to the CSIR NET exam pattern for maths.

Does access outlast your exam?

Pass: access ends at least a month after your exam date, which leaves a revision month. This is our recommendation. Fail: access that ends before the exam.

Our Complex Analysis Master Course is taught by SB Sir (Sunil Bansal) in Hinglish and costs ₹2,499 as of October 2026. It is built for CSIR NET and GATE. SB Sir secured All India Rank 11 in CSIR NET in December 2017 and is GATE MA qualified. He teaches pure mathematics at SBTech Math.

The course covers analytic functions, Cauchy-Riemann equations, complex integration, Liouville's theorem, power series, singularities, residues, the Schwarz lemma, the argument principle and bilinear transformations, another name for Möbius maps. The platform provides quizzes and live tests to practise on. Those topics sit in blocks 1 to 4, 6, 7 (the argument principle) and 8.

Run the checks above on our course too. Try a demo class first, and ask us where Taylor and Laurent series and Rouché's theorem are taught and whether the tests come with worked solutions. You should expect clear answers before you pay.

To buy, add the course to your cart, check out, and find it under My Courses. Registration needs your name and email address. We do not refund digital products, though we will consider a serious health issue straight after purchase, so use the demo classes first. Support runs from 9am to 7pm, Monday to Saturday.

If you are unsure whether you need one subject or a full CSIR NET and GATE course, tell us your exam and attempt on our contact page, and we will point you to the right one.