Advanced Complex Analysis: Residues, Conformal Maps, and Entire Functions
Advanced complex analysis goes far beyond the basics of complex arithmetic. It encompasses the deep theory of holomorphic and meromorphic functions, powerful integration techniques via the residue theorem, conformal mappings of domains, the remarkable behavior of entire functions, and analytic continuation — with applications across physics, engineering, and number theory.
Learning Objectives
After mastering this material, you will be able to:
- 1Verify holomorphicity using the Cauchy-Riemann equations and identify harmonic conjugates
- 2Apply the Cauchy integral formula and its generalizations to evaluate complex integrals
- 3Classify isolated singularities and compute Laurent series expansions
- 4Use the residue theorem to evaluate challenging real definite integrals via contour integration
- 5Construct conformal mappings between domains using Mobius transformations and the Schwarz-Christoffel formula
- 6State and apply the argument principle, Rouche's theorem, and the open mapping theorem
- 7Distinguish between Liouville's theorem, Picard's little theorem, and Picard's big theorem
- 8Perform analytic continuation and understand when global continuation is obstructed
- 9Connect complex analysis to harmonic analysis through the Poisson integral and Dirichlet problem
- 10Describe applications to fluid dynamics, electrostatics, signal processing, and the Riemann zeta function
1. Holomorphic Functions and the Cauchy-Riemann Equations
A function f : U to C defined on an open set U in C is holomorphic (or analytic) at a point z0 if its complex derivative exists there. This is a much stronger condition than real differentiability — requiring the limit of (f(z) - f(z0)) divided by (z - z0) to exist as z approaches z0 from any direction in the complex plane.
Complex Differentiability
Write f(z) = u(x, y) + i v(x, y) where z = x + iy and u, v are real-valued. The complex derivative requires the same limit regardless of how z approaches z0. Taking the limit along the real and imaginary axes and equating them yields the Cauchy-Riemann equations.
Cauchy-Riemann Equations
- du/dx = dv/dy
- du/dy = -dv/dx
If u and v have continuous first partial derivatives on U and satisfy the CR equations, then f = u + iv is holomorphic on U. The complex derivative is then f'(z) = du/dx + i dv/dx.
Geometric Interpretation
The Cauchy-Riemann equations encode the fact that the Jacobian matrix of (u, v) with respect to (x, y) is a rotation-and-scaling matrix — it has the form of multiplication by a complex number. This means holomorphic functions are conformal (angle-preserving) at points where the derivative is nonzero.
Harmonic Functions
If f = u + iv is holomorphic, then both u and v satisfy Laplace's equation: the Laplacian of u equals zero, and the Laplacian of v equals zero. Such functions are called harmonic. The function v is called the harmonic conjugate of u, and it is determined up to a constant. Every harmonic function on a simply connected domain is the real part of a holomorphic function.
Example: f(z) = z squared
- f(z) = (x + iy) squared = (x squared - y squared) + i(2xy)
- u(x,y) = x squared - y squared, v(x,y) = 2xy
- du/dx = 2x = dv/dy, du/dy = -2y = -dv/dx
- Laplacian of u = 2 - 2 = 0 (harmonic)
Key Theorem: Holomorphic Implies Infinitely Differentiable
Fundamental Difference from Real Analysis
In real analysis, a function can be differentiable once but not twice. In complex analysis, once differentiable implies infinitely differentiable and even representable by a convergent power series (analytic). This remarkable rigidity is a consequence of the Cauchy integral formula.
2. Cauchy's Theorem and the Integral Formula
Cauchy's theorem is the engine of complex analysis. It says that integrals of holomorphic functions around closed curves in simply connected domains always vanish — a fact with no analogue in real calculus.
Simply Connected Domains and Winding Numbers
A domain (connected open set) is simply connected if every closed curve in it can be continuously shrunk to a point — informally, it has no holes. The winding number of a closed curve gamma around a point z0 not on gamma counts (with sign) how many times gamma wraps around z0. For curves in simply connected domains, the winding number around any interior point is 0 or plus-or-minus 1.
Cauchy's Theorem
Statement
If f is holomorphic on a simply connected domain U, and gamma is any closed curve in U, then the contour integral of f(z) dz over gamma equals zero.
This means path independence holds: the integral of a holomorphic function between two points depends only on the endpoints, not the path taken — as long as the path stays in a simply connected region. This is the complex analogue of conservative vector fields in real calculus.
Cauchy Integral Formula
Cauchy Integral Formula and Its Derivatives
- f(z0) = (1 / 2*pi*i) * integral over gamma of f(z) / (z - z0) dz
- f'(z0) = (1 / 2*pi*i) * integral over gamma of f(z) / (z - z0) squared dz
- f to the n-th power (z0) = (n! / 2*pi*i) * integral of f(z) / (z - z0) to the (n+1) dz
The Cauchy integral formula is extraordinary: the values of a holomorphic function everywhere inside a circle are completely determined by its values on the boundary. This is a strong form of the principle that holomorphic functions are extremely rigid.
Consequences of Cauchy's Formula
Mean Value Property
The value of a holomorphic function at the center of a disk equals the average of its values on the boundary circle. This also holds for harmonic functions.
Maximum Modulus Principle
A non-constant holomorphic function on a domain cannot attain its maximum modulus at an interior point. The maximum is achieved only on the boundary of any compact subdomain.
Morera's Theorem
Converse of Cauchy's theorem: if f is continuous on an open set U and the integral of f over every triangle in U is zero, then f is holomorphic on U.
Liouville's Theorem
A bounded entire function (holomorphic on all of C) must be constant. This immediately implies the fundamental theorem of algebra: every non-constant polynomial has a complex root.
3. Taylor and Laurent Series
Power series are the natural language of complex analysis. Every holomorphic function is locally equal to its Taylor series, and Laurent series extend this to functions with isolated singularities.
Taylor Series and Radius of Convergence
If f is holomorphic on a disk D(z0, R) of radius R centered at z0, then f has a Taylor series that converges to f throughout the disk:
f(z) = sum from n=0 to infinity of [f to the n (z0) / n!] * (z - z0) to the n
The radius of convergence R is the distance from z0 to the nearest singularity of f. The series converges absolutely and uniformly on every closed subdisk of radius less than R.
Laurent Series
For a function holomorphic on an annulus r less than |z - z0| less than R (possibly with a singularity at z0), the Laurent series generalizes the Taylor series to include negative powers:
f(z) = sum from n=-infinity to infinity of a_n * (z - z0) to the n
where a_n = (1 / 2*pi*i) * integral of f(z) / (z - z0) to the (n+1) dz
The part with negative powers is the principal part; the part with non-negative powers is the analytic part. The coefficient a_(-1) is the residue of f at z0.
Classification of Isolated Singularities
Removable Singularity
Laurent series has no negative-power terms (principal part is zero). The function can be defined at z0 to make it holomorphic. Example: sin(z)/z at z = 0 — the Laurent series is 1 - z squared/6 + ...
Pole of Order n
Laurent series has finitely many negative-power terms, the lowest being a_(-n) / (z - z0) to the n with a_(-n) nonzero. Near a pole, |f(z)| goes to infinity. Example: 1/z squared has a pole of order 2 at z = 0.
Essential Singularity
Laurent series has infinitely many negative-power terms. By the Casorati-Weierstrass theorem, near an essential singularity the function is dense in C. Example: e to the (1/z) at z = 0.
Computing Laurent Series: Examples
Example: e to the (1/z) at z = 0
Using the Taylor series for e to the w = 1 + w + w squared/2! + ... with w = 1/z:
e to the (1/z) = 1 + 1/z + 1/(2! * z squared) + 1/(3! * z cubed) + ...
This has infinitely many negative powers, confirming an essential singularity. The residue (coefficient of 1/z) is 1.
4. The Residue Theorem and Contour Integration
The residue theorem is the culmination of the Cauchy theory. It converts contour integrals into algebraic computations of residues, enabling the evaluation of integrals that are intractable by real methods.
Statement of the Residue Theorem
Residue Theorem
Integral over gamma of f(z) dz = 2*pi*i * sum of Res(f, z_k)
where gamma is a positively oriented simple closed curve and the sum is over all poles z_k of f inside gamma.
Computing Residues
Simple Pole at z0
Res(f, z0) = lim as z goes to z0 of (z - z0) * f(z)
If f = g/h where g(z0) nonzero and h has a simple zero at z0: Res = g(z0) / h'(z0)
Pole of Order n at z0
Res(f, z0) = (1/(n-1)!) * lim of d to the (n-1) / dz to the (n-1) of [(z-z0) to the n * f(z)]
For n = 2: Res = lim of d/dz [(z - z0) squared * f(z)] as z goes to z0
Evaluating Real Integrals via Contour Integration
The residue theorem transforms complex contour integrals into finite sums. The strategy for real integrals is to embed the real line integral into a closed complex contour, show that unwanted parts of the contour contribute zero, then apply the residue theorem.
Type 1: Rational Functions Integrated from -infinity to infinity
Close the contour with a large semicircle in the upper half-plane. By Jordan's lemma, the semicircular arc contributes 0 as the radius grows, provided the degree of the denominator exceeds that of the numerator by at least 2. The real integral equals 2*pi*i times the sum of residues at poles in the upper half-plane.
Example: integral from -inf to inf of 1/(x squared + 1) dx
Pole at z = i in upper half-plane: Res = 1/(2i)
Result: 2*pi*i * (1/(2i)) = pi
Type 2: Trigonometric Integrals over [0, 2*pi]
Substitute z = e to the (i*theta), so cos(theta) = (z + 1/z)/2, sin(theta) = (z - 1/z)/(2i), and d(theta) = dz/(i*z). The integral over [0, 2*pi] becomes a contour integral over the unit circle, to which the residue theorem applies directly.
Type 3: Keyhole Contour for Fractional Powers
For integrals involving x to the (alpha - 1) times a rational function of x, use a keyhole contour that avoids the branch cut along the positive real axis. The contributions from the two sides of the cut differ by a factor of e to the (2*pi*i*alpha), giving a computable expression for the real integral.
Example: integral from 0 to inf of x to the (alpha-1) / (1+x) dx = pi / sin(pi*alpha)
Exam Tip: Residue Checklist
- 1. Identify all singularities and classify them (pole order?)
- 2. Determine which singularities lie inside your contour
- 3. Use the correct residue formula for each pole order
- 4. Verify that arc contributions vanish (Jordan's lemma or direct estimate)
- 5. Account for orientation: clockwise contour gives minus sign
5. Conformal Mappings
A conformal mapping is a holomorphic function that preserves angles locally. At every point where the derivative is nonzero, the function acts as a rotation and scaling — so curves meeting at angle theta still meet at angle theta under the map. Conformal mappings are the fundamental tool for transforming complicated domains into simple ones.
Mobius Transformations
A Mobius transformation (linear fractional transformation) has the form f(z) = (az + b) / (cz + d) where ad - bc is nonzero. These are the conformal automorphisms of the Riemann sphere.
Key Properties of Mobius Transformations
- •They map circles and lines to circles and lines (circles on the Riemann sphere to circles)
- •They are determined by their values at three distinct points (three-point normalization)
- •They form a group under composition, isomorphic to PSL(2, C)
- •The unique Mobius transformation sending z1, z2, z3 to 0, 1, infinity is the cross-ratio
Standard Mobius Maps to Know
- Upper half-plane to unit disk: f(z) = (z - i) / (z + i)
- Unit disk to unit disk fixing 0: f(z) = e to the (i*theta) * z
- Unit disk to itself, 0 maps to a: f(z) = (z - a) / (1 - a_bar * z)
The Riemann Mapping Theorem
Theorem Statement
Every simply connected proper open subset U of C is conformally equivalent to the open unit disk D. That is, there exists a biholomorphic map f : U to D. The map is unique if we normalize by requiring f(z0) = 0 and f'(z0) greater than 0 for a chosen z0 in U.
The Riemann mapping theorem is existential — it does not provide an explicit formula for arbitrary domains. For domains bounded by polygons, the Schwarz-Christoffel formula gives an explicit conformal map from the upper half-plane to the polygon.
Schwarz-Christoffel Formula
Schwarz-Christoffel Map (Upper Half-Plane to Polygon)
f(z) = A + C * integral of product over k of (zeta - x_k) to the (alpha_k - 1) d(zeta)
Here x_1 less than x_2 less than ... less than x_n are the real preimages of the polygon vertices, and alpha_k * pi are the interior angles at those vertices. The formula maps the upper half-plane conformally onto the interior of a polygon.
The Schwarz Lemma
If f maps the unit disk to itself holomorphically with f(0) = 0, then |f(z)| is at most |z| for all z in the disk, and |f'(0)| is at most 1. Equality holds if and only if f is a rotation: f(z) = e to the (i*theta) times z. The Schwarz lemma is proved elegantly by applying the maximum modulus principle to f(z)/z.
6. Meromorphic Functions: Argument Principle and Rouche's Theorem
A meromorphic function on a domain is holomorphic except at isolated poles. Meromorphic functions include all rational functions and many transcendental functions. The argument principle connects the geometry of how a function winds around zero with the algebraic count of zeros and poles.
The Argument Principle
Statement
(1 / 2*pi*i) * integral over gamma of f'(z)/f(z) dz = N - P
where N is the number of zeros of f inside gamma (counted with multiplicity) and P is the number of poles inside gamma (counted with order). The left side equals the winding number of the curve f(gamma) around the origin.
The argument principle is a bridge between topology (winding numbers) and algebra (counting zeros and poles). It underlies Nyquist stability analysis in control theory and the study of zeros of the Riemann zeta function.
Rouche's Theorem
Statement
If f and g are holomorphic inside and on a simple closed curve gamma, and |g(z)| is strictly less than |f(z)| for all z on gamma, then f and f + g have the same number of zeros inside gamma.
Rouche's theorem is the standard tool for counting zeros in a region. It gives a new proof of the fundamental theorem of algebra: for p(z) = z to the n plus lower terms, take f(z) = z to the n and g = the lower-order terms. On a large enough circle, |f| dominates |g|, so p has exactly n zeros inside.
Open Mapping Theorem and Inverse Function Theorem
A non-constant holomorphic function maps open sets to open sets. This is a consequence of the argument principle and has no analogue for real differentiable functions (which can map open sets to single points). Combined with the argument principle, it implies that if f'(z0) is nonzero, then f is locally invertible near z0 and the inverse is holomorphic.
Practice Problem: Count the Zeros
How many zeros does p(z) = z to the 5 minus 3z + 1 have inside the unit disk?
Solution:
Apply Rouche's theorem with f(z) = -3z and g(z) = z to the 5 + 1. On the unit circle |z| = 1: |f(z)| = 3 and |g(z)| is at most |z to the 5| + 1 = 2. Since 2 less than 3, f + g = p(z) - (p(z) - f(z) - g(z)) -- actually: write p = f + g where f = -3z (3 zeros inside? No, -3z has one zero). Alternatively: f = -3z has 1 zero inside |z| less than 1, and |g| = |z to the 5 + 1| is at most 2 less than 3 = |f|. So p has exactly 1 zero inside the unit disk.
7. Entire Functions: Liouville, Picard, and Weierstrass
An entire function is holomorphic on all of C. The class of entire functions is extremely rich — it includes polynomials, exponentials, sine, cosine, and the Bessel functions. Several deep theorems describe the range and structure of entire functions.
Liouville's Theorem and Its Consequences
Liouville's Theorem
Every bounded entire function is constant. Equivalently, a non-constant entire function is unbounded.
Proof sketch: If |f(z)| is at most M everywhere, Cauchy's integral formula gives |f'(z0)| at most M/R for any R. Sending R to infinity gives f'(z0) = 0 for all z0, so f is constant.
Picard's Little Theorem
Statement
A non-constant entire function takes every complex value with at most one exception. That is, there is at most one value that the function never attains.
The exponential function e to the z takes every value except 0 — this is the one exceptional value. No entire function can omit two or more values (unless it is constant).
Picard's Big Theorem
Statement
In any punctured disk around an essential singularity, a holomorphic function takes every complex value with at most one exception infinitely often.
For e to the (1/z) at z = 0: in any neighborhood of 0, this function takes every nonzero complex value infinitely often. It omits only 0. The Casorati-Weierstrass theorem gives the weaker statement that the image is dense in C.
Weierstrass Factorization Theorem
Just as polynomials factor into linear factors (z - z_k) over C, entire functions can be factored in terms of their zeros. The Weierstrass factorization theorem says that given any sequence of complex numbers z_1, z_2, ... with |z_n| going to infinity, there exists an entire function with exactly those zeros.
Weierstrass Products
f(z) = z to the m * e to the g(z) * product over n of E_p_n(z/z_n)
where g is an entire function and E_p are elementary factors ensuring convergence. Example: sin(pi*z) = pi*z * product over n nonzero of (1 - z squared/n squared). This is the Hadamard factorization for functions of finite order.
Order of Entire Functions
The order of an entire function f is defined as the infimum of all rho greater than 0 such that |f(z)| is at most e to the |z| to the rho for large |z|. Polynomials have order 0, e to the z has order 1, e to the (e to the z) is not of finite order. Hadamard's theorem connects the order to the exponent of convergence of the zero sequence.
8. Analytic Continuation
Analytic continuation is the process of extending a holomorphic function defined on one domain to a larger domain, using the rigidity of holomorphic functions. The identity principle guarantees that if two holomorphic functions agree on any set with an accumulation point, they agree everywhere on their common domain.
Direct Continuation and Identity Principle
If f is holomorphic on a disk D1 and g is holomorphic on an overlapping disk D2, and f = g on D1 intersect D2, then g is the unique continuation of f to D2. The continuation is performed by chaining overlapping disks, creating an analytic function element at each step.
Identity Principle
If f and g are holomorphic on a connected domain U, and the set where f equals g has an accumulation point in U, then f equals g everywhere on U. In particular, a holomorphic function that vanishes on any convergent sequence of distinct points must be identically zero.
Monodromy Theorem
Statement
If a function element can be continued analytically along every path in a simply connected domain U, then these continuations are consistent: the result depends only on the endpoint, not the path. Thus the function has a well-defined single-valued extension to all of U.
When the domain is not simply connected, continuation along different paths may yield different values — this is the phenomenon of multivaluedness. The Riemann surface is the correct domain for a multivalued function.
Branch Cuts and Riemann Surfaces
Functions like log(z) and z to the (1/2) are multivalued in C because continuing around the origin gives a different value upon return. A branch cut is a curve (typically along the negative real axis) that is removed from the domain to make the function single-valued. The Riemann surface is a multi-sheeted complex manifold on which the function becomes single-valued globally.
Principal Branch of log(z)
- log(z) = ln|z| + i*arg(z), where arg(z) is in (-pi, pi]
- Branch cut: negative real axis (z less than 0)
- Two continuations around origin: differ by 2*pi*i
- Riemann surface: infinitely many sheets connected at 0
Application: The Riemann Zeta Function
The Riemann zeta function is initially defined for Re(s) greater than 1 by the series sum of 1/n to the s. Analytic continuation extends it to all of C except for a simple pole at s = 1. The functional equation connects zeta(s) with zeta(1 - s). The Riemann hypothesis — that all non-trivial zeros have real part 1/2 — is the most famous open problem in mathematics.
9. Harmonic Analysis: Poisson Integral and the Dirichlet Problem
The deep connection between holomorphic and harmonic functions allows complex analysis to solve classical problems in mathematical physics. The Dirichlet problem asks for a harmonic function on a domain with prescribed boundary values. For the disk, the solution is given by the Poisson integral formula.
The Poisson Integral Formula
Poisson Formula for the Disk
u(r, theta) = (1 / 2*pi) * integral from 0 to 2*pi of P(r, theta - phi) * f(phi) d(phi)
where P(r, theta) = (1 - r squared) / (1 - 2r cos(theta) + r squared)
This gives the unique harmonic function u on the unit disk with boundary values f. The Poisson kernel P(r, theta) is an approximate identity as r approaches 1 from below.
Boundary Behavior and Fatou's Theorem
Fatou's theorem states that if u is a bounded harmonic function on the unit disk, then for almost every boundary point (in the sense of Lebesgue measure), the non-tangential limit of u at that point exists. This is a fundamental result about the boundary behavior of functions in the Hardy spaces.
Hardy Spaces
The Hardy space H to the p consists of holomorphic functions on the disk whose p-th power means over circles are uniformly bounded. Hardy spaces are the natural function spaces for complex analysis and provide the framework connecting complex analysis to Fourier analysis. The study of H to the 2 is essentially the theory of power series with square-summable coefficients.
Schwarz Reflection Principle
Reflection Principle
If f is holomorphic on the upper half-plane, continuous up to a portion of the real axis, and real-valued there, then f extends to a holomorphic function on the symmetric domain by setting f(z_bar) = f_bar(z) for z in the lower half-plane. This is a powerful tool for extending functions across boundaries and is used in conformal mapping and the study of the Riemann zeta functional equation.
10. Applications to Physics, Engineering, and Number Theory
Complex analysis is not merely an abstract discipline — it is indispensable across physics, engineering, and mathematics. The techniques of conformal mapping, contour integration, and potential theory solve concrete problems in fluid dynamics, electrostatics, heat flow, signal processing, and analytic number theory.
Fluid Dynamics: Potential Flow
For irrotational, incompressible (potential) flow in 2D, the velocity field derives from a potential function phi satisfying Laplace's equation. The complex potential w(z) = phi + i*psi (where psi is the stream function) is holomorphic. Conformal mappings transform complicated geometries (flow around an airfoil) to simple ones (flow around a cylinder), solving the Joukowski airfoil problem and providing the foundation for thin airfoil theory.
Electrostatics and Heat Flow
Electric potential in 2D satisfies Laplace's equation, as does temperature in steady-state heat flow. The real and imaginary parts of a holomorphic function give conjugate harmonic functions, providing pairs of equipotential lines and field lines. Conformal maps let engineers solve Laplace's equation on complicated electrode geometries by mapping them to simpler ones where the solution is known.
Signal Processing and the z-Transform
The z-transform of a discrete-time signal is a power series evaluated at a complex number z. Its region of convergence is an annulus in the complex plane. Pole-zero analysis of transfer functions uses the argument principle and Nyquist stability criterion (based on the winding number of the frequency response around -1). Contour integration recovers the inverse z-transform via the residue theorem.
Number Theory: The Riemann Zeta Function
The Riemann zeta function zeta(s) encodes the distribution of prime numbers through the Euler product formula. The prime number theorem — pi(x) is asymptotic to x/ln(x) — follows from the fact that zeta(s) has no zeros on the line Re(s) = 1. This is proved using contour integration and the analytic continuation of zeta. The Riemann hypothesis would give the sharpest possible bounds on the error in the prime number theorem.
Control Theory: Nyquist Stability
A feedback control system with open-loop transfer function G(s)H(s) is stable if and only if the Nyquist plot of G(i*omega)H(i*omega) as omega ranges over the real line winds around -1 in the correct way. This is precisely the argument principle applied to the function 1 + G(s)H(s): the system is stable iff this function has no zeros in the right half-plane, which is read off from the winding number. Complex analysis thus provides the mathematical foundation for feedback design in aerospace, automotive, and electrical engineering.
Practice Problems with Solutions
Determine where f(z) = |z| squared is complex differentiable. Is it holomorphic anywhere?
Show Solution
Write f(z) = x squared + y squared, so u = x squared + y squared and v = 0.
du/dx = 2x, dv/dy = 0: requires 2x = 0, so x = 0
du/dy = 2y, -dv/dx = 0: requires 2y = 0, so y = 0
The CR equations are satisfied only at z = 0. So f is complex differentiable only at z = 0, with f'(0) = 0. It is not holomorphic anywhere, since there is no neighborhood of 0 on which it is differentiable (holomorphicity requires differentiability in an open set).
Compute the residue of f(z) = e to the z divided by (z squared (z - 1)) at each pole, and evaluate the integral of f around the circle |z| = 2.
Show Solution
Poles: z = 0 (order 2) and z = 1 (simple). Both lie inside |z| = 2.
Res at z=1: lim (z-1) * e to the z / (z squared (z-1)) = e to the 1 / 1 = e
Res at z=0 (order 2): d/dz [z squared * e to the z / (z squared (z-1))] = d/dz [e to the z / (z-1)]
= [e to the z (z-1) - e to the z] / (z-1) squared, at z=0: [(-1) - 1] / 1 = -2
Res(f, 0) = -2
Integral = 2*pi*i * (Res at 0 + Res at 1) = 2*pi*i * (-2 + e) = 2*pi*i*(e - 2)
Evaluate the integral from 0 to infinity of x squared / (x to the 4 + 1) dx using contour integration.
Show Solution
Since the integrand is even, the integral from 0 to inf equals (1/2) times the integral from -inf to inf.
Poles of z squared / (z to the 4 + 1): at z to the 4 = -1, i.e., z = e to the (i*pi*(2k+1)/4) for k = 0, 1, 2, 3.
Upper half-plane poles: z1 = e to the (i*pi/4) = (1+i)/sqrt(2) and z2 = e to the (i*3*pi/4) = (-1+i)/sqrt(2)
Res at z_k of z squared / (z to the 4 + 1) = z_k squared / (4 * z_k cubed) = 1 / (4 * z_k)
Sum of residues = 1/(4 z1) + 1/(4 z2) = (1/4)(e to the (-i*pi/4) + e to the (-i*3*pi/4))
= (1/4)[(1-i)/sqrt(2) + (-1-i)/sqrt(2)] = (1/4)[-2i/sqrt(2)] = -i/(2*sqrt(2))
Full integral = 2*pi*i * (-i/(2*sqrt(2))) = pi/sqrt(2)
Answer: (1/2) * pi/sqrt(2) = pi / (2*sqrt(2)) = pi*sqrt(2)/4
Find a conformal map sending the upper half-plane to the unit disk that maps i to 0.
Show Solution
Use the standard Mobius transformation. The map f(z) = (z - i)/(z + i) sends the upper half-plane to the unit disk.
Check: f(i) = (i - i)/(i + i) = 0/2i = 0. Correct.
For real z = x: |f(x)| = |x - i|/|x + i| = sqrt(x squared + 1) / sqrt(x squared + 1) = 1. So the real axis maps to the unit circle.
Since the upper half-plane is simply connected and maps continuously into the disk with the boundary mapping to the boundary, by the open mapping theorem this is the desired conformal equivalence. Normalize further: f'(i) = [2i]/(2i) squared ... but the basic map is f(z) = (z - i)/(z + i).
Classify the singularity of f(z) = sin(z) / z cubed at z = 0 and compute the residue.
Show Solution
Expand sin(z) in its Taylor series: sin(z) = z - z cubed/6 + z to the 5/120 - ...
sin(z)/z cubed = 1/z squared - 1/6 + z squared/120 - ...
The principal part is 1/z squared (one negative-power term with n = -2). This is a pole of order 2 at z = 0.
The residue is the coefficient of 1/z, which is 0.
Note: the coefficient of 1/z squared is 1 (so it is a second-order pole), but there is no 1/z term, so Res(f, 0) = 0. The integral of f around any small circle centered at 0 is 0.
Show that all roots of p(z) = z to the 7 - 5z cubed + 2 lie inside the disk |z| less than 2.
Show Solution
On the circle |z| = 2, apply Rouche with f(z) = z to the 7 and g(z) = -5z cubed + 2.
|f(z)| = 2 to the 7 = 128
|g(z)| is at most 5|z| cubed + 2 = 5 * 8 + 2 = 42
Since 42 less than 128, by Rouche's theorem p = f + g has the same number of zeros inside |z| = 2 as f(z) = z to the 7, which has exactly 7 zeros at the origin (counting multiplicity).
Therefore all 7 roots of p lie inside the disk |z| less than 2.
Exam Tips and Common Mistakes
Cauchy-Riemann Equations
- !CR equations are necessary for complex differentiability but not sufficient alone — you also need the partial derivatives to be continuous (or use Looman-Menchoff theorem).
- !Satisfying CR at a single point does not make f holomorphic — holomorphic means differentiable in an open set.
Residues and Contour Integration
- !Always verify which poles are inside your contour before summing residues — a common error is including poles on the boundary or in the wrong half-plane.
- !For poles on the contour (typically at the boundary of the real line), use an indented contour with a small semicircle. Poles on the contour contribute half their residue (for simple poles on the real axis with a semicircular indent).
- !Jordan's lemma requires |f(z)| to go to 0 uniformly as |z| goes to infinity on the semicircle. Verify this before claiming the arc integral vanishes.
Laurent Series and Singularity Classification
- !The Laurent series is valid in an annulus — specify the region of validity when computing (e.g., 0 less than |z| less than 1 vs |z| greater than 1).
- !Poles and removable singularities are called non-essential isolated singularities. At a pole, |f(z)| goes to infinity. At a removable singularity, f remains bounded.
Theorems to Know Cold
Liouville
Bounded entire implies constant
Picard Little
Entire omits at most one value
Picard Big
Near essential sing., misses at most one value
Rouche
|g| less than |f| on boundary implies same zeros
Argument Principle
Integral of f'/f = 2*pi*i*(N - P)
Riemann Mapping
Simply connected proper subset ≅ unit disk
Monodromy
Simple connected domain implies single-valued continuation
Maximum Modulus
Maximum on boundary, not interior
Related Topics
Complex Analysis (Foundations)
Start here for complex numbers, basic holomorphicity, and elementary contour integration before tackling advanced topics.
Precalculus Study Guide
Build the algebraic and trigonometric foundations needed for calculus and ultimately complex analysis.
Real Analysis
The rigorous foundations of calculus, completeness, uniform convergence, and power series that underpin complex analysis.
Functional Analysis
Hardy spaces, operator theory on Hilbert spaces, and spectral theory — the natural continuation of advanced complex analysis.
Quick Reference: Key Results
| Result | Statement | Key Use |
|---|---|---|
| Cauchy's Theorem | Holomorphic on simply connected: integral around any loop = 0 | Path independence, primitive existence |
| Cauchy Integral Formula | f(z0) = (1/2*pi*i) * integral of f(z)/(z-z0) dz | Values from boundary data, Taylor coefficients |
| Residue Theorem | Integral = 2*pi*i * sum of residues | Evaluating contour and real integrals |
| Laurent Series | f(z) = sum of a_n (z-z0) to the n, -inf to inf | Classifying singularities, computing residues |
| Argument Principle | (1/2*pi*i) * integral of f'/f = N - P | Counting zeros and poles |
| Rouche's Theorem | |g| less than |f| on boundary implies zeros of f+g = zeros of f | Locating zeros of polynomials |
| Liouville's Theorem | Bounded entire implies constant | Fundamental theorem of algebra |
| Riemann Mapping Thm | Every simply connected proper subset ≅ disk | Conformal equivalence |
| Picard Little | Entire omits at most one value | Range of entire functions |
| Weierstrass Factorization | Entire function = exponential times infinite product over zeros | Constructing functions with prescribed zeros |
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