Complex analysis begins with a simple enlargement of the number line, then rapidly changes what “function”, “angle”, “distance”, “infinity” and “differentiability” can mean.
The complex plane turns a number z=x+iy into a geometric point (x,y). Algebra and geometry become two descriptions of the same object. Multiplication combines scaling and rotation; roots distribute themselves around circles; exponential notation converts trigonometric structure into algebra; and rational functions become geometric transformations of the extended plane.
This guide is the entry to BTT’s Complex Analysis route. It owns the plane, basic complex functions, limits, continuity, mappings and the Riemann sphere. Holomorphic differentiation belongs to R26.02, contour integration and residues to R26.03, and analytic continuation and branches to R26.04.
Prerequisites: algebra, trigonometry, functions, vectors, sequences and limits. For a rigorous limit foundation, begin with Limits, Continuity and Sequences.
Reading route: arithmetic → geometry → modulus and argument → polar form → exponential form → roots and powers → functions → elementary mappings → complex limits → continuity → extended plane → stereographic projection → Möbius transformations → theorem boundaries → practice → solutions.
1. The complex plane identifies algebra with geometry
Write z=x+iy with x,y∈R and i²=−1. The real part is Re z=x and the imaginary part is Im z=y.
We identify z with the point (x,y) in R². Addition becomes vector addition:
(x+iy)+(u+iv)=(x+u)+i(y+v).
This identification is not cosmetic. Complex multiplication will add a rotational structure not present in ordinary vector addition.
2. Complex multiplication mixes scale and angle
(x+iy)(u+iv)=(xu−yv)+i(xv+yu).
In Cartesian coordinates this looks like a bilinear formula. In polar form it becomes much more revealing: moduli multiply and arguments add.
This is why the complex plane is naturally adapted to rotations, oscillations and periodic phenomena.
3. Conjugation reflects across the real axis
The conjugate of z=x+iy is z̄=x−iy.
Geometrically, conjugation reflects the point across the real axis.
Important identities include z+z̄=2Re z, z−z̄=2i Im z, and zz̄=|z|².
4. Modulus is Euclidean distance from the origin
The modulus is
|z|=√(x²+y²).
It satisfies the triangle inequality |z+w|≤|z|+|w| and multiplicativity |zw|=|z||w|.
Distance in C is inherited from the Euclidean plane: d(z,w)=|z−w|.
5. Argument records direction but is multivalued
For z≠0, an argument θ satisfies z=|z|(cosθ+i sinθ).
If θ is one argument, so is θ+2πk for every integer k.
Thus “the argument” is naturally a set of values modulo 2π. A principal argument chooses one interval, often (−π,π], but that choice introduces a discontinuity along a branch cut.
6. Polar form exposes multiplicative structure
For z≠0, write
z=r(cosθ+i sinθ), r>0.
If z₁=r₁(cosθ₁+i sinθ₁) and z₂=r₂(cosθ₂+i sinθ₂), then
z₁z₂=r₁r₂[cos(θ₁+θ₂)+i sin(θ₁+θ₂)].
Multiplication scales by r₂ and rotates by θ₂.
7. Euler’s formula compresses rotation into exponentials
Euler’s identity
e^{iθ}=cosθ+i sinθ
lets us write z=re^{iθ}. Then multiplication becomes
r₁e^{iθ₁}·r₂e^{iθ₂}=r₁r₂e^{i(θ₁+θ₂)}.
The exponential notation turns geometric rotation into additive angle arithmetic.
8. De Moivre’s formula handles powers
For integer n,
[r(cosθ+i sinθ)]^n=r^n[cos(nθ)+i sin(nθ)].
This follows immediately from repeated multiplication or Euler notation.
It explains why powers multiply angles and why roots later divide angles.
9. nth roots form regular polygons
To solve z^n=w with w=Re^{iφ}, write
z_k=R^{1/n}e^{i(φ+2πk)/n}, k=0,…,n−1.
The n roots have equal modulus and equally spaced arguments. They form the vertices of a regular n-gon centred at the origin.
10. Worked example: fourth roots of 16
Write 16=16e^{i2πm}. The fourth roots have modulus 2 and arguments mπ/2.
Hence the roots are 2, 2i, −2, −2i.
The geometry makes completeness visible: every fourth root lies on the circle |z|=2, separated by right angles.
11. Regions in C are described by modulus and argument inequalities
|z−a|<r is an open disk centred at a. |z−a|=r is a circle. Re z>c is a half-plane. α<arg(z−a)<β is a sector.
Reading such sets geometrically is an essential complex-analysis skill because domains determine which branches and theorems are legal later.
12. A complex function maps plane geometry to plane geometry
A function f:D⊂C→C can be written
f(z)=u(x,y)+iv(x,y).
Thus a complex function is simultaneously one complex-valued rule and two coupled real-valued functions u and v.
The coupling becomes decisive in R26.02, where complex differentiability forces strong relations between u and v.
13. Translation, rotation and scaling are elementary mappings
f(z)=z+a translates every point by a.
f(z)=cz with c=ρe^{iφ} scales distances from the origin by ρ and rotates angles by φ.
These maps preserve straight lines and circles in predictable ways and form the simplest local models for more general holomorphic maps.
14. Squaring doubles argument and squares modulus
If z=re^{iθ}, then z²=r²e^{i2θ}.
A sector of angle α becomes one of angle 2α, provided the map remains one-to-one on the chosen sector.
Globally z² is two-to-one away from zero because z and −z have the same square.
15. The reciprocal map inverts radius and reverses angle
For z=re^{iθ}≠0,
1/z=r^{-1}e^{-iθ}.
Large radii become small and positive angles become negative. The reciprocal map naturally exchanges zero and infinity once the plane is extended to the Riemann sphere.
16. The complex exponential is periodic
For z=x+iy,
e^z=e^x(cos y+i sin y).
Therefore e^{z+2πi}=e^z.
The map converts vertical translation by 2πi into the same output, so it cannot have a single-valued global inverse on C\{0}. This is the structural origin of the multivalued complex logarithm.
17. Complex sine and cosine are defined by exponentials
Define
cos z=(e^{iz}+e^{-iz})/2, sin z=(e^{iz}−e^{-iz})/(2i).
For real z these recover ordinary trigonometric functions, but for complex inputs they are unbounded in new directions because hyperbolic growth appears.
18. Complex limits use ordinary Euclidean distance
We say lim_{z→z₀}f(z)=L if for every ε>0 there exists δ>0 such that
0<|z−z₀|<δ implies |f(z)−L|<ε.
This is the same metric definition as in R². The difference is that a complex point can be approached from infinitely many directions, so path independence becomes crucial.
19. Path disagreement disproves a complex limit
Consider f(z)=z̄/z for z≠0.
Along the real axis, z=x gives z̄/z=1. Along the imaginary axis, z=iy gives z̄/z=(−iy)/(iy)=−1.
The limit at 0 does not exist.
As in multivariable calculus, finitely many agreeing paths never prove existence; disagreement is only a disproof tool.
20. Algebraic limit laws still require existence
If f→L and g→M, then f+g→L+M, fg→LM, and f/g→L/M when M≠0.
Polynomials are continuous everywhere. Rational functions are continuous wherever their denominators are nonzero.
These statements use the same epsilon algebra as real analysis, with modulus replacing absolute value.
21. Continuity means limits agree with values
f is continuous at z₀ if
lim_{z→z₀}f(z)=f(z₀).
Equivalently, for every ε>0 there is δ>0 such that |z−z₀|<δ implies |f(z)−f(z₀)|<ε.
Continuity is far weaker than holomorphicity. A function can be continuous everywhere and nowhere complex differentiable.
22. Conjugation is continuous but not holomorphic
The map f(z)=z̄ is an isometry: |z̄−w̄|=|z−w|, so it is continuous everywhere.
Yet its difference quotient at any point depends on the direction of approach, so it is nowhere complex differentiable.
This is the first warning that complex differentiability is dramatically stronger than real differentiability in R².
23. Infinity can be added as one geometric point
The extended complex plane is
Ĉ=C∪{∞}.
Sequences z_n tend to ∞ when |z_n|→∞.
A function f(z) tends to L as z→∞ if f(1/w)→L as w→0. This substitution converts behaviour at infinity into ordinary local analysis near zero.
24. The Riemann sphere compactifies the complex plane
Geometrically, Ĉ can be identified with a sphere via stereographic projection. The north pole represents ∞.
This compactification is powerful because rational functions become maps from the sphere to itself. Poles are no longer mysterious blow-ups; they are points mapped to ∞.
25. Stereographic projection gives an explicit coordinate bridge
Project a point on the unit sphere, other than the north pole, along the line through the north pole onto the equatorial plane.
The resulting plane coordinate is a complex number z. Reversing the construction sends every finite z back to a unique sphere point.
The missing north pole is exactly the extra point ∞.
26. Möbius transformations act naturally on the sphere
A Möbius transformation has the form
T(z)=(az+b)/(cz+d), ad−bc≠0.
It extends to Ĉ by sending −d/c to ∞ when c≠0 and ∞ to a/c.
Two matrices differing by a nonzero scalar define the same Möbius transformation.
27. Möbius maps preserve generalized circles
A generalized circle means either an ordinary circle or a straight line, viewed as a circle through ∞ on the Riemann sphere.
Möbius transformations send generalized circles to generalized circles.
This makes them fundamental tools for moving complicated domains into simpler standard regions such as disks and half-planes.
28. Three distinct points determine a unique Möbius map
Given distinct z₁,z₂,z₃ and distinct w₁,w₂,w₃ on Ĉ, there is a unique Möbius transformation with T(z_j)=w_j.
One efficient construction uses cross-ratios.
This rigidity distinguishes Möbius geometry from arbitrary plane transformations.
29. Cross-ratio is a Möbius invariant
For suitable distinct points, define
[z₁,z₂;z₃,z₄]=((z₁−z₃)(z₂−z₄))/((z₁−z₄)(z₂−z₃)).
Möbius transformations preserve this quantity.
Four points lie on one generalized circle exactly when their cross-ratio is real, with suitable ordering conventions.
30. Worked Möbius map: half-plane to disk
The Cayley transform
T(z)=(z−i)/(z+i)
maps the upper half-plane Im z>0 onto the unit disk |w|<1.
The real axis maps to the unit circle. The point z=i maps to 0, while ∞ maps to 1.
This is a prototype for conformal normalisation developed later in R26.05.
31. Common theorem-boundary errors
- A principal argument is not globally continuous on C\{0}.
- Agreement along several paths does not prove a multivariable or complex limit.
- Continuity does not imply complex differentiability.
- z↦z² is not globally one-to-one.
- e^z has no global single-valued inverse on C\{0} because of 2πi-periodicity.
- A pole at infinity must be analysed on the extended plane, not by informal “very large z” language alone.
- Möbius transformations require ad−bc≠0; otherwise numerator and denominator are linearly dependent.
32. A dependable complex-plane workflow
Choose Cartesian form for addition, real/imaginary parts and direct coordinate limits. Choose polar or exponential form for multiplication, powers, roots and rotation.
When a function involves argument, logarithm or roots, state the domain and branch convention before manipulating formulas.
When studying a limit, remember that approach occurs through the entire plane. When studying infinity, move to w=1/z and analyse near zero.
When a rational transformation appears, consider the Riemann sphere and Möbius geometry rather than treating poles as exceptional algebraic accidents.
33. Independent practice: twenty questions
- For z=3−4i, find Re z, Im z, z̄ and |z|.
- Express −1+i√3 in polar form.
- Multiply 2e^{iπ/3} and 3e^{−iπ/6}.
- Use De Moivre to compute (cosθ+i sinθ)^5.
- Find all cube roots of 8.
- Describe geometrically |z−2i|<3.
- Describe Re z≤−1.
- What does multiplication by −i do geometrically?
- Show e^{z+2πi}=e^z.
- Write sin z in exponential form.
- State the epsilon–delta definition of a complex limit.
- Use two paths to show z̄/z has no limit at 0.
- Why is conjugation continuous?
- Why is continuity weaker than holomorphicity?
- Define convergence z_n→∞ on the extended plane.
- What point on the Riemann sphere corresponds to infinity?
- State the general Möbius transformation.
- What condition prevents a Möbius transformation from degenerating?
- What geometric objects do Möbius maps preserve?
- Where does T(z)=(z−i)/(z+i) send i and ∞?
34. Worked solutions and checks
1. Re z=3, Im z=−4, z̄=3+4i, |z|=5.
2. Modulus 2 and principal argument 2π/3, so 2e^{i2π/3}.
3. 6e^{iπ/6}.
4. cos5θ+i sin5θ.
5. 2, 2e^{i2π/3}, 2e^{i4π/3}.
6. Open disk radius 3 centred at 2i.
7. Closed half-plane on and to the left of x=−1.
8. Rotation by −π/2 with no change of modulus.
9. e^{z+2πi}=e^z e^{2πi}=e^z.
10. sin z=(e^{iz}−e^{-iz})/(2i).
11. For every ε>0 there exists δ>0 such that 0<|z−z₀|<δ implies |f(z)−L|<ε.
12. Along z=x the quotient is 1; along z=iy it is −1.
13. |z̄−w̄|=|z−w|, so conjugation is an isometry.
14. Continuity only controls output values near inputs; complex differentiability requires one direction-independent complex linear derivative.
15. z_n→∞ exactly when |z_n|→∞.
16. The north pole under stereographic projection.
17. T(z)=(az+b)/(cz+d).
18. ad−bc≠0.
19. Generalized circles: circles and lines.
20. T(i)=0 and T(∞)=1.
35. Where this guide hands off
R26.02 asks what happens when the complex difference quotient has the same limit from every direction. The answer is surprisingly rigid: Cauchy–Riemann relations appear, local power series emerge, harmonic functions enter, and a small amount of complex differentiability produces far more structure than ordinary real differentiability.
Sources and further study
For a standard university route, see MIT OpenCourseWare 18.04 Complex Variables with Applications. This BTT article uses original exposition and worked examples while keeping school/H2 complex-number pages and quantum complex-amplitude pages as separate application owners.