Complex Functions Practice Quiz with a Step-by-Step Interactive Lesson
Use the quiz at the top of the page to practice complex functions and core complex analysis ideas with the most important definitions and tests: complex numbers \(z=x+iy\) and complex conjugate \(\overline{z}\), modulus \(|z|\) and argument \(\arg z\), Euler's formula \(e^{i\theta}=\cos\theta+i\sin\theta\) and polar form \(z=re^{i\theta}\), analytic / holomorphic functions and the Cauchy-Riemann equations, entire functions (holomorphic on \(\mathbb{C}\)), complex exponentials and mappings like \(w=e^z\) and \(w=\tfrac{1}{z}\), singularities (removable, poles, essential), Laurent series intuition, residues and quick residue computations, and basic contour integrals such as \(\oint z^n\,dz\). If you want a refresher, click Start lesson to open a step-by-step guide with worked examples and quick checks.
How this complex functions practice works
1. Take the quiz: answer the complex numbers and complex functions questions at the top of the page.
2. Open the lesson (optional): review conjugates, modulus/argument, analyticity, mappings, singularities, residues, and contour integrals with clear examples.
3. Retry: return to the quiz and apply the complex analysis rules immediately.
What you will learn in the complex functions lesson
Complex numbers, modulus, argument, and conjugates
Rectangular form \(z=x+iy\) and basic arithmetic
Complex conjugate \(\overline{z}=x-iy\) and identities like \(z\overline{z}=|z|^2\)
Modulus \(|z|=\sqrt{x^2+y^2}\) and argument \(\arg z\) for polar form
Complex exponential, polar form, and mappings
Euler's formula \(e^{i\theta}=\cos\theta+i\sin\theta\) and \(z=re^{i\theta}\)
Exponential map \(w=e^z\): periodicity \(e^{z+2\pi i}=e^z\) and images of lines
Reciprocal map \(w=\tfrac{1}{z}\): circles/lines mapping and inversion geometry
Holomorphic and analytic functions
Complex differentiability and the meaning of holomorphic / analytic
Cauchy-Riemann equations for \(f(z)=u(x,y)+iv(x,y)\)
Common checks: why \(f(z)=\overline{z}\) and \(f(z)=|z|^2\) are not analytic
Singularities, residues, and contour integrals
Removable singularities vs. poles vs. essential singularities
Residue at a simple pole and fast computation for rational functions
Core fact: \(\displaystyle \oint_{|z|=1} z^n\,dz = 0\) for all integers \(n≠ -1\)
Back to the quiz
When you are ready, return to the quiz at the top of the page and keep practicing complex functions and complex analysis.
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Complex Functions
Step-by-step guide
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Complex Functions Lesson
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Lesson Overview
What you will learn
Purpose: Build a clear understanding of complex numbers and complex functions so you can compute with \(z=x+iy\), use conjugates, modulus, and argument, apply Euler’s formula and the complex exponential, decide when a function is holomorphic / analytic (via the Cauchy-Riemann equations), classify singularities (removable, poles, essential), compute residues, and evaluate simple contour integrals.
Success criteria
Compute with complex numbers and simplify powers like \((1+i)^3\).
Use the complex conjugate \(\overline{z}\) and compute \(|z|\) correctly.
Convert between rectangular form \(x+iy\) and polar form \(re^{i\theta}\).
Use Euler’s formula \(e^{i\theta}=\cos\theta+i\sin\theta\) and understand \(e^z\).
Understand the mapping \(w=e^z\) (periodicity and images of lines).
Decide when a function is holomorphic and why \(\overline{z}\) and \(|z|^2\) are not analytic.
Know what entire means and identify common entire functions.
Classify singularities (removable vs. pole vs. essential singularity).
Compute the residue at a simple pole for rational functions.
Evaluate basic contour integrals like \(\oint_{|z|=1} z^n\,dz\).
Key vocabulary
Complex number: \(z=x+iy\), where \(i^2=-1\).
Conjugate: \(\overline{z}=x-iy\).
Modulus: \(|z|=\sqrt{x^2+y^2}\).
Argument: \(\arg z\) is an angle \(\theta\) with \(z=re^{i\theta}\) (multi-valued, principal value often used).
Holomorphic (analytic): complex differentiable on an open set.
Entire: holomorphic on all of \(\mathbb{C}\).
Singularity: a point where a function fails to be holomorphic.
Residue: the coefficient of \((z-a)^{-1}\) in the Laurent expansion around \(a\).
Quick pre-check
Pre-check 1: What is the complex conjugate of \(2-5i\)?
Hint: Conjugation flips the sign of the imaginary part.
Pre-check 2: Under \(w=e^z\), what is the image of the real axis \(z=x\)?
Hint: If \(z=x\) then \(w=e^x\), which is real and positive.
Complex Number Basics
Complex numbers, conjugates, modulus, and fast algebra
Learning goal: Compute confidently with complex numbers and use conjugates and modulus to simplify expressions.
Key idea
A complex number is \(z=x+iy\), where \(x=\Re(z)\) and \(y=\Im(z)\). The complex conjugate is \[ \overline{z}=x-iy. \] The modulus (absolute value) is \[ |z|=\sqrt{x^2+y^2}. \] A key identity is \[ z\overline{z}=|z|^2. \] This is why multiplying by the conjugate helps simplify fractions like \(\frac{1}{a+bi}\).
Worked example
Example: Compute \((1+i)^3\).
First square: \[ (1+i)^2 = 1+2i+i^2 = 2i. \] Then multiply by \((1+i)\): \[ (1+i)^3=(1+i)\cdot(2i)=2i+2i^2=2i-2=-2+2i. \]
Learning goal: Use \(e^z\) and basic conformal mapping intuition to track how lines and circles transform.
Key idea
Euler’s formula connects exponentials and trigonometry: \[ e^{i\theta}=\cos\theta+i\sin\theta. \] For a general complex number \(z=x+iy\), \[ e^z=e^{x+iy}=e^x e^{iy}=e^x(\cos y+i\sin y). \] This shows two important mapping effects:
Changing \(x\) scales the magnitude: \(|e^z|=e^x\).
Changing \(y\) rotates the argument by \(y\) (mod \(2\pi\)).
Also, the exponential is periodic in the imaginary direction: \[ e^{z+2\pi i}=e^z. \] So \(e^z\) is not injective on \(\mathbb{C}\).
Worked example
Example: Under \(w=e^z\), what is the image of the real axis \(z=x\)?
If \(z=x\) then \(w=e^x\), which is real and positive. So the real axis maps to the positive real axis \((0,\infty)\).
Try it
Try it 1: Is \(f(z)=e^z\) injective on \(\mathbb{C}\)?
Hint: If two different inputs differ by \(2\pi i\), they have the same output.
Try it 2: Under the map \(w=\tfrac{1}{z}\), what does the unit circle \(|z|=1\) map to?
Hint: If \(|z|=1\), then \(|1/z|=1/|z|=1\).
Summary
\(e^{x+iy}=e^x(\cos y+i\sin y)\): scale by \(e^x\), rotate by \(y\).
\(e^z\) is not injective on \(\mathbb{C}\) because of \(2\pi i\)-periodicity.
Inversion \(w=1/z\) sends \(|z|=1\) to \(|w|=1\).
Holomorphic & Cauchy-Riemann
Analytic functions, Cauchy-Riemann equations, and common non-examples
Learning goal: Decide whether a function is holomorphic (analytic) and recognize the most common traps: \(\overline{z}\), \(|z|^2\), and multi-valued powers.
Key idea
Write a complex function as \[ f(z)=u(x,y)+iv(x,y),\quad z=x+iy. \] If \(u\) and \(v\) have continuous first partial derivatives and \(f\) is complex differentiable, then the Cauchy-Riemann equations must hold: \[ u_x=v_y,\qquad u_y=-v_x. \] Many functions fail to be holomorphic because they depend on \(\overline{z}\) or \(|z|\), which mixes \(x\) and \(y\) in a non-analytic way.
Worked example
Example: Is \(f(z)=\overline{z}\) holomorphic anywhere?
Write \(f(z)=\overline{z}=x-iy\). Then \(u(x,y)=x\) and \(v(x,y)=-y\). Compute partials: \[ u_x=1,\; u_y=0,\; v_x=0,\; v_y=-1. \] Cauchy-Riemann would require \(u_x=v_y\), i.e. \(1=-1\), which is impossible. So \(f(z)=\overline{z}\) is not holomorphic anywhere on \(\mathbb{C}\).
Try it
Try it 1: Is \(f(z)=|z|^2\) analytic?
Hint: \(|z|^2=x^2+y^2\). Try Cauchy-Riemann: it cannot hold on any open set.
Try it 2: Is \(f(z)=z^z\) analytic on \(\mathbb{C}\)?
Hint: Usually \(z^z=e^{z\Log z}\). A single-valued \(\Log z\) needs a branch cut, so you cannot define it holomorphically on all of \(\mathbb{C}\).
\(\overline{z}\) and \(|z|^2\) are not holomorphic anywhere on \(\mathbb{C}\).
Multi-valued functions (like \(z^z\)) require branch choices; they are not analytic on all of \(\mathbb{C}\).
Entire Functions & Series
Entire functions, power series, and geometric series in \(\mathbb{C}\)
Learning goal: Recognize entire functions and use standard convergence facts for complex power series.
Key idea
A function is entire if it is holomorphic on all of \(\mathbb{C}\). Classic examples include polynomials and \(e^z\), \(\sin z\), \(\cos z\). Power series in the complex plane behave like real power series: they have a radius of convergence \(R\), and they converge absolutely for \(|z-a|<R\) and diverge for \(|z-a|>R\).
A key example is the geometric series: \[ \sum_{n=0}^{\infty}(z-a)^n \] which converges exactly when \(|z-a|<1\) and then sums to \(\frac{1}{1-(z-a)}\).
Worked example
Example: For which \(z\) does \(\sum_{n=0}^{\infty}(z-1)^n\) converge?
This is a geometric series with ratio \((z-1)\). It converges when \(|z-1|<1\), i.e. inside the open disk centered at \(1\) with radius \(1\).
Try it
Try it 1: Which of these is entire?
Hint: \(\sin z\) is holomorphic everywhere; rational functions have poles; \(\overline{z}\) is not holomorphic.
Try it 2: For which \(z\) does the series \(\sum_{n=0}^\infty (z-1)^n\) converge?
Hint: A geometric series \(\sum r^n\) converges exactly when \(|r|<1\).
Summary
Entire means holomorphic on all of \(\mathbb{C}\).
\(e^z,\sin z,\cos z\), and polynomials are entire.
\(\sum_{n=0}^{\infty}(z-1)^n\) converges exactly when \(|z-1|<1\).
Singularities
Classifying singularities: removable, poles, and essential
Learning goal: Recognize and classify isolated singularities quickly using expansions and key examples.
Key idea
An isolated singularity at \(z=a\) is a point where \(f\) is not holomorphic at \(a\) but is holomorphic on a punctured neighborhood \(0<|z-a|<r\). Three main types:
Removable singularity: \(f\) can be redefined at \(a\) to become holomorphic (often happens when a factor cancels).
Pole of order \(m\): \((z-a)^m f(z)\) is holomorphic and nonzero at \(a\). A simple pole is \(m=1\).
Essential singularity: neither removable nor a pole; Laurent series has infinitely many negative-power terms.
Worked example
Example: What type of singularity does \(f(z)=e^{1/z}\) have at \(z=0\)?
The Laurent series is \[ e^{1/z}=\sum_{n=0}^{\infty}\frac{1}{n!}\,z^{-n}. \] This has infinitely many negative powers, so \(z=0\) is an essential singularity.
Try it
Try it 1: What type of singularity does \(f(z)=\dfrac{\sin z}{z}\) have at \(z=0\)?
Hint: \(\sin z = z - \frac{z^3}{3!}+\cdots\). Divide by \(z\) to see the limit at 0.
Try it 2: What type of singularity does \(f(z)=\dfrac{\sin(1/z)}{z}\) have at \(z=0\)?
Hint: \(\sin(1/z)\) already has an essential singularity at 0; multiplying by \(1/z\) cannot turn it into a pole or removable singularity.
Summary
\(e^{1/z}\) and \(\sin(1/z)\) have essential singularities at \(z=0\).
\(\frac{\sin z}{z}\) has a removable singularity at \(0\) (define value as \(1\)).
\(\frac{\sin(1/z)}{z}\) has an essential singularity at \(0\).
Residues & Poles
Residues at simple poles and fast computations
Learning goal: Compute residues quickly for simple rational functions and recognize simple poles.
Key idea
If \(f\) has a simple pole at \(z=a\), then its residue is \[ \operatorname{Res}(f,a)=\lim_{z\to a}(z-a)f(z). \] A rational function \(\frac{p(z)}{q(z)}\) typically has poles where \(q(z)=0\) (if \(p(a)≠ 0\) and the zero is simple).
Worked example
Example: What is the residue of \(f(z)=\dfrac{3z}{z-1}\) at \(z=1\)?
This is a simple pole at \(z=1\). Compute: \[ \operatorname{Res}(f,1)=\lim_{z\to 1}(z-1)\frac{3z}{z-1}=\lim_{z\to 1}3z=3. \]
Try it
Try it 1: Which of these is a simple pole of \(\tan z\)?
Hint: \(\tan z=\frac{\sin z}{\cos z}\). Poles occur where \(\cos z=0\).
Try it 2: What is the residue of \(f(z)=\dfrac{3z}{z-1}\) at \(z=1\)?
Hint: For a simple pole at \(a\), use \(\lim_{z\to a}(z-a)f(z)\).
\(\tan z\) has simple poles at \(\frac{\pi}{2}+k\pi\).
Contour Integrals & Big Picture
Contour integrals, angle preservation, and final recap
Learning goal: Use core contour integral facts and connect them to analyticity and conformal mapping.
Core contour integral fact
For the unit circle \(|z|=1\), a fundamental identity is: \[ \oint_{|z|=1} z^n\,dz = \begin{cases} 2\pi i, & n=-1,\\ 0, & n≠ -1, \end{cases} \] where \(n\) is an integer. In particular, \[ \oint_{|z|=1} z^2\,dz = 0. \]
Angle preservation (conformality)
Holomorphic functions with nonzero derivative preserve angles between smooth curves (they are conformal). The map \(f(z)=\overline{z}\) is not holomorphic; it reflects the plane and does not preserve oriented angles, so it is not conformal.
The function \(z^2\) is holomorphic everywhere, and it has an antiderivative \(\frac{z^3}{3}\). The integral of a holomorphic derivative around a closed curve is \(0\), so \[ \oint_{|z|=1} z^2\,dz = 0. \]
Try it
Try it 1: Does \(f(z)=\overline{z}\) preserve oriented angles (is it conformal)?
Hint: Conformal (angle-preserving) maps are holomorphic with nonzero derivative. Conjugation is not holomorphic.
Try it 2: What is \(\displaystyle \oint_{|z|=1} z^2 \,dz\)?
Hint: \(\oint z^n\,dz=0\) for \(n≠ -1\) around \(|z|=1\).
Final recap
Conjugate and modulus: \(\overline{x+iy}=x-iy\), \(|x+iy|=\sqrt{x^2+y^2}\), \(z\overline{z}=|z|^2\).
Exponential: \(e^{x+iy}=e^x(\cos y+i\sin y)\), and \(e^{z+2\pi i}=e^z\) (not injective on \(\mathbb{C}\)).
Holomorphic: check Cauchy-Riemann; \(\overline{z}\) and \(|z|^2\) are not holomorphic anywhere.
Entire: holomorphic on all of \(\mathbb{C}\); examples include \(e^z\), \(\sin z\), \(\cos z\), polynomials.
Singularities: removable (e.g. \(\sin z/z\) at \(0\)), poles, essential (e.g. \(e^{1/z}\) at \(0\)).
Residue at a simple pole: \(\operatorname{Res}(f,a)=\lim_{z\to a}(z-a)f(z)\).
Contour integral: \(\oint_{|z|=1} z^2\,dz=0\).
Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the complex function skill you need: algebra, mappings, analyticity, singularities, residues, or integrals.
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