Each worksheet prints cleanly on A4 or Letter. Answer keys are on a separate page.

The Lumenary · Advanced undergraduate

Complex analysis

Name: ______________________

Date: ______________________

You will learn to

Test complex differentiability with the Cauchy–Riemann equations, and explain why $z \mapsto \bar z$ is not holomorphic although it is real-differentiable
Compute contour integrals, prove Goursat's theorem and Cauchy's theorem in a disc, and prove Cauchy's integral formula
Deduce that holomorphic functions are infinitely differentiable and analytic, with the Cauchy estimates, Liouville's theorem and the fundamental theorem of algebra
Use Laurent series and the residue theorem to evaluate real integrals
Explain conformality, isolated singularities, and why $1/z$ has no holomorphic extension to $0$

1.
Write $z = x + iy$ . Which of these functions are holomorphic on all of $\C$ (entire)? Choose all that apply.
- A. $z^3 + 2z$
- B. $\bar z = x - iy$
- C. $|z|^2 = x^2 + y^2$
- D. $e^z = e^x(\cos y + i \sin y)$
- E. $x^2 - y^2 + 2ixy$
2.
Compute $\displaystyle \frac{1}{2\pi i} \oint_{|z| = 2} \frac{3z^2 + 5}{z - 1}\,dz$ (circle traversed once anticlockwise).
3.
For which real constants $a, b$ is $f(x + iy) = x^2 + a y^2 + i\,bxy$ holomorphic on $\C$ ?
- A. $a = -1$ , $b = 2$
- B. $a = 1$ , $b = 2$
- C. $a = -1$ , $b = -2$
- D.No values work
4.
How many zeros (counted with multiplicity) does $p(z) = z^7 - 5z^3 + 1$ have in the open unit disc $|z| < 1$ ?
5.
Liouville's theorem: a bounded holomorphic function on all of $\C$ is constant. Which example shows that "on all of $\C$ " cannot be weakened?
- A. $f(z) = z$ on the open unit disc
- B. $f(z) = \sin z$ on $\C$
- C. $f(z) = 1/z$ on $\C \setminus \{0\}$
- D. $f(z) = e^{-z^2}$ on $\C$
6.
Which Lean statement says that complex conjugation is nowhere complex-differentiable?
- A.∀ z : ℂ, ¬ DifferentiableAt ℂ (starRingEnd ℂ) z
- B.¬ ∀ z : ℂ, DifferentiableAt ℂ (starRingEnd ℂ) z
- C.∀ z : ℂ, ¬ DifferentiableAt ℝ (starRingEnd ℂ) z
- D.∀ z : ℂ, deriv (starRingEnd ℂ) z = 0
7.
Use the residue theorem on the upper semicircle to evaluate $\displaystyle \int_{-\infty}^{\infty} \frac{\cos 2x}{1 + x^2}\,dx$ .
- A. $\pi / e^2$
- B. $\pi / e$
- C. $\pi / (2e^2)$
- D. $\pi \cosh 2 / e^2$
8.
Write winding_number(points, p) for a closed polygon given by the list points of complex vertices (the last vertex joins back to the first) and a complex point p not on it. Add up the change in the argument of $z - p$ along each edge — the angle cmath.phase((b - p) / (a - p)) from vertex a to vertex b, which lies in $(-\pi, \pi]$ — and divide the total by $2\pi$ . Return the nearest integer. (This is $\frac{1}{2\pi i}\oint \frac{dz}{z - p}$ , computed exactly edge by edge.)
Computational — Practice
9.
A student argues: " $\frac{d}{dz}\log z = \frac1z$ , so $\log z$ is a primitive of $\frac1z$ on $\C \setminus \{0\}$ , and therefore $\oint_{|z|=1} \frac{dz}{z} = \log(1) - \log(1) = 0$ ." But the integral is $2\pi i$ . What is wrong?
- A.There is no holomorphic function $\log z$ on all of $\C \setminus \{0\}$ : every branch is discontinuous along some ray from $0$ , so it is not a primitive along the whole circle.
- B.The derivative of $\log z$ is not $1/z$ .
- C.Cauchy's theorem does not apply because $1/z$ is not continuous on the circle.
- D.The circle must be parametrised clockwise.
10.
Which statements are true? Choose all that apply.
- A.If $f$ is holomorphic on the open unit disc and $f = 0$ on the segment $[0, \tfrac12]$ , then $f = 0$ on the whole disc.
- B.Every entire function is a polynomial.
- C.If $f$ is holomorphic on $0 < |z| < 1$ and bounded there, then $f$ extends holomorphically to $|z| < 1$ .
- D.If $f : \R \to \R$ is infinitely differentiable, it equals its Taylor series near every point.

Answer key — Complex analysis

1.
Entire: $z^3 + 2z$ , $e^z$ and $x^2 - y^2 + 2ixy = z^2$ . The function $\bar z$ fails the Cauchy–Riemann equations everywhere, and $|z|^2 = z \bar z$ satisfies them only at $0$ , so it is complex-differentiable at $0$ but holomorphic nowhere.
2.
$f(z) = 3z^2 + 5$ is entire and $|1| < 2$ , so Cauchy's integral formula gives $\frac{1}{2\pi i}\oint \frac{f(z)}{z - 1}\,dz = f(1) = 8$ . (Equivalently: the residue of the integrand at its only pole $z = 1$ is $f(1) = 8$ .)
3.
With $u = x^2 + ay^2$ , $v = bxy$ : $u_x = 2x$ , $v_y = bx$ , so $b = 2$ ; $u_y = 2ay$ , $-v_x = -by$ , so $2a = -b = -2$ and $a = -1$ . The partial derivatives are continuous, so $f = x^2 - y^2 + 2ixy = z^2$ is entire.
4.
Take $f(z) = -5z^3$ and $g(z) = z^7 + 1$ . On $|z| = 1$ , $|g| \le 2 < 5 = |f|$ , and $p = f + g$ has no zeros on the circle (since $|p| \ge 5 - 2 > 0$ ). By Rouché, $p$ has as many zeros in $|z| < 1$ as $-5z^3$ , namely $3$ (a triple zero at $0$ ).
5.
$f(z) = z$ on the unit disc is bounded and holomorphic but not constant. Liouville's proof needs Cauchy's estimate $|f'(c)| \le M/R$ for arbitrarily large $R$ . Surprisingly, removing a single point from $\C$ is not enough to create a counterexample, by Riemann's removable singularity theorem.
6.
∀ z : ℂ, ¬ DifferentiableAt ℂ (starRingEnd ℂ) z. The negated universal statement is weaker; the ℝ-version is false (conjugation is ℝ-linear); and the deriv statement is true only because Mathlib sets deriv f z = 0 when f is not differentiable at z, so it carries no information.
7.
Let $f(z) = \frac{e^{2iz}}{1 + z^2}$ . On the upper semicircle of radius $R > 1$ , $|f| \le \frac{1}{R^2 - 1}$ , so its contribution is at most $\frac{\pi R}{R^2 - 1} \to 0$ . The only pole inside is $z = i$ , with residue $\frac{e^{2i \cdot i}}{2i} = \frac{e^{-2}}{2i}$ . Hence $\int_{-\infty}^\infty \frac{e^{2ix}}{1 + x^2}\,dx = \pi e^{-2}$ , and taking real parts gives $\int \frac{\cos 2x}{1 + x^2}\,dx = \pi/e^2$ .
8.
```
import cmath
import math

def winding_number(points, p):
    total = 0.0
    n = len(points)
    for k in range(n):
        a = points[k]
        b = points[(k + 1) % n]
        total += cmath.phase((b - p) / (a - p))
    return round(total / (2 * math.pi))
```
Along a straight edge that does not pass through $p$ , the argument of $z - p$ changes by less than $\pi$ in absolute value, so phase returns exactly that change. The total change around a closed curve is a multiple of $2\pi$ — the winding number times $2\pi$ . This is $\operatorname{Im} \oint \frac{dz}{z - p}$ ; the real part, the change in $\log|z - p|$ , is $0$ around a closed curve.
9.
A primitive must be a single-valued holomorphic function on an open set containing the whole circle. Any continuous choice of $\arg z$ on the circle minus a point jumps by $2\pi$ across that point, so $\log z$ has a jump of $2\pi i$ and is not a primitive along the full circle. In fact the integral $2\pi i$ proves that no primitive of $1/z$ exists on $\C \setminus \{0\}$ .
10.
True: the identity theorem and Riemann's removable singularity theorem. False: $e^z$ is entire but not a polynomial; and $e^{-1/x^2}$ is smooth on $\R$ but not equal to its (zero) Taylor series at $0$ . That last example shows how much stronger complex differentiability is than real smoothness.