# Complex analysis: a contour is part of the hypothesis — Study worksheet

[Study guides](https://unireadinglist.com/guides/complex-analysis/)

[Course topic: Complex Analysis](https://unireadinglist.com/subjects/mathematics/#course-complex-analysis)

Sources checked: 2026-11-04

Study route and exercise: University Reading List editorial project.

A first route through holomorphic functions, Cauchy formulas and isolated singularities. Beck and collaborators provide a free first-course spine; Ahlfors is a more compressed classical treatment; Needham adds geometric explanation. This is not a complete course in topology or advanced complex function theory.

## Choose your starting point

- [ ] I need a free first course
  Start with Beck and collaborators, building integration and the residue theorem in sequence.

- [ ] I want a formal classical treatment
  Choose Ahlfors after proof-based calculus; expect to supply intermediate arguments.

- [ ] I need a geometric reason for the formulas
  Use Needham alongside an analytic spine, drawing domains and deformations.

## Compare the three books

### A First Course in Complex Analysis

Matthias Beck; Gerald Marchesi; Dennis Pixton; Lucas Sabalka · 1.6 · 2025
Free first-course spine

Choose it when: You can differentiate and integrate real functions and write short proofs.
Use another route when: You need only a quick table of integration tricks.
Selected reading: Complex integration, Cauchy formulas, singularities and residues in version 1.6.
Limits and trade-offs: The sequence develops needed ideas rather than assuming a substantial real-analysis course, but proofs still require work.

#### Try before choosing

For e^z/\[z²(z−3)\], mark all poles and the circle |z|=1 before choosing a theorem.
Check your work: Only 0 lies inside; e^z/(z−3) is analytic on and inside that circle.

Your notes:

________________________________

ISBN: 9781944325206

[Official free reading](https://matthbeck.github.io/complex.html)

[Book record and access options](https://unireadinglist.com/subjects/mathematics/a-first-course-in-complex-analysis/)

### Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable

Lars V. Ahlfors · 3 · 2021
Compressed analytic reference

Choose it when: You are comfortable with limits, series and proof and want precise hypotheses.
Use another route when: A first encounter with proofs requires many small scaffolded steps.
Selected reading: Analytic functions, complex integration and local behavior at singularities in the selected third-edition reprint.
Limits and trade-offs: Concise arguments repay rereading; a reprint date does not make it a newly rewritten course.

#### Try before choosing

Differentiate e^z/(z−3) at zero and keep both product-rule terms.
Check your work: The derivative is −1/3−1/9=−4/9, not just −1/3.

Your notes:

________________________________

ISBN: 9781470467678

[Book record and access options](https://unireadinglist.com/subjects/mathematics/complex-analysis-ahlfors/)

### Visual Complex Analysis

Tristan Needham · 25th anniversary · 2023
Geometric companion

Choose it when: Pictures of winding and deformation help you understand an integral.
Use another route when: You require a short computational reference or a replacement for checking analytic domains.
Selected reading: Complex Integration: Cauchy’s Theorem and Cauchy’s Formula, with the earlier geometric foundations.
Limits and trade-offs: Visual explanation is substantial, but the drawing must still identify every singularity crossed.

#### Try before choosing

Draw a deformation from radius 1 to radius 4 and mark the obstruction.
Check your work: The pole at 3 lies between the circles, so deformation invariance cannot be invoked across it.

Your notes:

________________________________

ISBN: 9780192868923

[Book record and access options](https://unireadinglist.com/subjects/mathematics/visual-complex-analysis/)

## Check your preparation

### Can I differentiate a quotient of complex analytic functions?

Your notes:

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### What does “analytic inside” include?

Your notes:

________________________________

### Can I distinguish an orientation from a radius?

Your notes:

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## A six-step study plan

- [ ] Map the domain

Selected reading: Complex functions and isolated singularities.

What to do: Draw both circles and classify the two poles.

Check your work: Zero is double and 3 is simple; neither lies on either path.

Your notes:

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- [ ] State the usable theorem

Selected reading: Cauchy derivative formula.

What to do: Write f=F/z² and specify where F is analytic.

Check your work: The hypothesis holds on the small disk and fails on the large disk.

Your notes:

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- [ ] Compute the local coefficient

Selected reading: Local series or derivative formula.

What to do: Find the coefficient of z^−1 at zero using two methods.

Check your work: Both give −4/9.

Your notes:

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- [ ] Solve the small contour

Selected reading: Cauchy formula or residues.

What to do: Give the oriented integral and explain which singularity contributes.

Check your work: The answer is −8πi/9.

Your notes:

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- [ ] Change the domain

Selected reading: Residue theorem and the follow-up.

What to do: Add the newly enclosed simple-pole residue.

Check your work: The additional contribution is 2πi e³/9.

Your notes:

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- [ ] Audit a deformation claim

Selected reading: Your diagrams and both solutions.

What to do: Write one valid deformation within the punctured domain and one invalid crossing.

Check your work: A theorem failure is explained by a specific pole, not by the circle being larger.

Your notes:

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## Worked exercise: A double pole inside a small circle

Compute the counterclockwise integral of f(z)=e^z/\[z²(z−3)\] around |z|=1, giving a domain justification.

Assumptions: Exact complex arithmetic; e^z is entire; each circle is traversed once counterclockwise. There are no singularities on the path.

Your notes:

________________________________

## Change an assumption: A new enclosed singularity changes the theorem route

Keep f and orientation, but integrate around |z|=4. Explain why applying the earlier single Cauchy-derivative formula over this whole disk is invalid.

Your notes:

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## Before moving on

- State the domain assumptions of a Cauchy formula.
- Compute a double-pole residue by derivative and series.
- Explain a changed contour using the new enclosed pole.

## Answer and interpretation

### Can I differentiate a quotient of complex analytic functions?

Away from z=3, F(z)=e^z/(z−3) has F′(z)=e^z/(z−3)−e^z/(z−3)².

### What does “analytic inside” include?

The whole enclosed domain needed by the selected theorem, not just points on the contour.

### Can I distinguish an orientation from a radius?

Both cases below are counterclockwise. Reversing orientation would separately negate each integral.

### A double pole inside a small circle

1. The singularities are a double pole at 0 and a simple pole at 3. Only 0 is inside |z|=1. Put F(z)=e^z/(z−3), analytic on a neighborhood of the closed unit disk.
2. Cauchy’s derivative formula gives the integral 2πi F′(0). Product differentiation gives F′(0)=−1/3−1/9=−4/9.
3. Equivalently, e^z=1+z+… and 1/(z−3)=−1/3−z/9−… near zero, so the z coefficient of F is −4/9 and the residue of f is −4/9.

The integral is −8πi/9.

Test the result: The derivative and local-series calculations agree. The pole at 3 contributes nothing to this contour because it lies outside, not because its residue is zero.

### Change an assumption: A new enclosed singularity changes the theorem route

1. Now both poles are enclosed. The residue at zero remains −4/9; local behavior does not depend on the chosen enclosing circle.
2. The residue at the simple pole 3 is lim(z→3)(z−3)f(z)=e³/9. The residue theorem therefore sums −4/9+e³/9.
3. F=e^z/(z−3) is not analytic on the larger disk. Shrinking that circle directly to the small one crosses 3, so the previous analytic-domain hypothesis fails.

The larger integral is 2πi(e³−4)/9; its difference from the small-circle integral is 2πi e³/9.

Test the result: A deformation that avoids both poles preserves the integral. One that crosses 3 changes the winding around that pole, accounting for exactly the computed difference.

## Common mistakes

### Only checking the path

A function can be finite on a circle and still have a pole inside it.

### Dropping a derivative term

Both the exponential and reciprocal factor contribute to F′.

### Treating deformation as an unrestricted drawing

The homotopy must stay in a domain where the integrand is analytic.

## Sources and edition checks

- [Authors: first course, version1.6](https://matthbeck.github.io/complex.html)
  The authors’ version 1.6 develops a first rigorous route to residues and is freely available under its stated licence.

- [CiNii: exact AMS third-edition reprint](https://ci.nii.ac.jp/ncid/BC09389490)
  The selected AMS item is a third-edition reprint, not a newly authored 2021 edition.

- [Oxford:25th-anniversary geometric approach](https://academic.oup.com/book/45765)
  The anniversary work provides geometric treatments of complex integration and Cauchy formulas.

These are proposed study activities, not official course assignments or a claim that every book has been read cover to cover.

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