Complex number Study Guide

📖 Core Concepts Imaginary unit: defined by \(i^{2} = -1\). Complex number: \(z = a + bi\) with real part \(\operatorname{Re}(z)=a\) and imaginary part \(\operatorname{Im}(z)=b\). Complex plane: horizontal axis = real part, vertical axis = imaginary part; \(a \leftrightarrow (a,0)\), \(bi \leftrightarrow (0,b)\). Field & vector space: \(\mathbb{C}\) is a field (every non‑zero element has an inverse) and a 2‑D real vector space with basis \(\{1,i\}\). Conjugate: \(\overline{z}=a-bi\). Modulus (absolute value): \(|z|=\sqrt{a^{2}+b^{2}}=\sqrt{z\overline{z}}\). Argument: \(\theta=\arg(z)\) satisfies \(z = |z|(\cos\theta + i\sin\theta)\); principal value \(\theta\in(-\pi,\pi]\). Polar / exponential form: \(z = r e^{i\theta}\) with \(r=|z|\). Euler’s formula \(e^{i\theta}= \cos\theta + i\sin\theta\). De Moivre: \((\cos\theta+i\sin\theta)^{n}= \cos(n\theta)+i\sin(n\theta)\); in polar form \(z^{n}=r^{n}e^{in\theta}\). \(n\)‑th roots: \(\displaystyle \sqrt[n]{z}= r^{1/n}\Bigl(\cos\frac{\theta+2k\pi}{n}+i\sin\frac{\theta+2k\pi}{n}\Bigr),\;k=0,\dots ,n-1\). Fundamental Theorem of Algebra: every non‑constant polynomial with complex coefficients has a complex root; \(\mathbb{C}\) is algebraically closed. Holomorphic (analytic) function: \(f=u+iv\) is holomorphic ⇔ Cauchy–Riemann equations hold: \[ \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y},\qquad \frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}. \] --- 📌 Must Remember Addition/Subtraction: add/subtract real parts and imaginary parts separately. Multiplication: \((a{1}+b{1}i)(a{2}+b{2}i) = (a{1}a{2}-b{1}b{2})+(a{1}b{2}+a{2}b{1})i\). Division: \(\displaystyle\frac{z{1}}{z{2}} = \frac{(a{1}+b{1}i)(a{2}-b{2}i)}{a{2}^{2}+b{2}^{2}}\). Reciprocal: \(\displaystyle \frac{1}{a+bi}= \frac{a-bi}{a^{2}+b^{2}}\). Polar multiplication/division: magnitudes multiply/divide, arguments add/subtract. Euler: \(e^{i\theta}= \cos\theta+i\sin\theta\). De Moivre: \(z^{n}=r^{n}(\cos n\theta+i\sin n\theta)\). \(i\) powers cycle: \(i^{0}=1,\;i^{1}=i,\;i^{2}=-1,\;i^{3}=-i,\;i^{4}=1,\dots\). Complex logarithm: \(\log w = \ln r + i(\theta+2k\pi)\), principal value restricts \(\theta\in(-\pi,\pi]\). Norm of Gaussian integer: \(N(x+iy)=x^{2}+y^{2}=z\overline{z}\). Sum‑of‑two‑squares criterion: a prime \(p\equiv3\pmod4\) must appear with an even exponent in the factorisation of \(n\) for \(n\) to be a sum of two squares. --- 🔄 Key Processes Add/Subtract: \[ (a{1}+b{1}i)\pm(a{2}+b{2}i)=(a{1}\pm a{2})+(b{1}\pm b{2})i. \] Multiply (Cartesian): use the formula above or FOIL, remembering \(i^{2}=-1\). Convert to polar: \(r=|z|=\sqrt{a^{2}+b^{2}}\). \(\theta=\operatorname{atan2}(b,a)\). Write \(z=r(\cos\theta+i\sin\theta)=re^{i\theta}\). Divide: multiply numerator and denominator by the conjugate of the denominator, then simplify. Power (integer \(n\)): switch to polar, apply De Moivre, convert back if needed. Root extraction: compute \(r^{1/n}\); generate \(n\) angles \(\frac{\theta+2k\pi}{n}\). Check holomorphic: compute partial derivatives of \(u\) and \(v\); verify Cauchy–Riemann equations. Logarithm: find \(r\) and \(\theta\); write \(\log w = \ln r + i\theta\) plus \(2k\pi i\) for other branches. --- 🔍 Key Comparisons Cartesian vs. Polar: Cartesian easy for addition/subtraction. Polar simplifies multiplication, division, powers, and roots (magnitudes multiply, arguments add). Conjugate vs. Reciprocal: Conjugate flips sign of imaginary part; reciprocal additionally scales by \(1/|z|^{2}\). Principal argument vs. General argument: Principal \(\in(-\pi,\pi]\); general adds integer multiples of \(2\pi\). \(i^{n}\) cycle vs. General exponentiation: Powers of \(i\) repeat every 4; for other bases use De Moivre. Gaussian integer norm vs. ordinary absolute value: Norm is integer‑valued (\(x^{2}+y^{2}\)); ordinary modulus may be irrational. --- ⚠️ Common Misunderstandings Argument range: forgetting the principal range and using a value outside \((-π,π]\) can give the wrong sign in polar form. Division step: omitting the conjugate multiplier leads to an incorrect denominator. Logarithm uniqueness: assuming \(\log w\) is single‑valued; ignore the \(2k\pi i\) family. Roots count: missing that there are exactly \(n\) distinct \(n\)‑th roots; sometimes only one is written. Cauchy–Riemann necessity: thinking it is sufficient without checking differentiability of partial derivatives. --- 🧠 Mental Models / Intuition Multiplication as geometry: multiply magnitudes (dilation) and add angles (rotation). Visualize \(z\) as an arrow; \(z{1}z{2}\) stretches by \(|z{1}|\) and turns by \(\arg(z{1})\). Euler’s circle: \(e^{i\theta}\) is a point on the unit circle; the real part is the x‑coordinate, the imaginary part the y‑coordinate. Norm multiplicativity: \(N(z{1}z{2}) = N(z{1})N(z{2})\) → product of distances equals distance of product; useful for sums of squares. De Moivre as angle scaling: raising to \(n\) multiplies the angle by \(n\); extracting roots divides the angle by \(n\). --- 🚩 Exceptions & Edge Cases Zero: \(\arg(0)\) undefined; division by zero impossible. Logarithm branches: each integer \(k\) gives a distinct value; principal log restricts \(k=0\). Roots when \(\theta\) on boundary: angles \(\theta = \pi\) or \(-\pi\) produce duplicate roots if not handled with the full \(2k\pi\) term. Holomorphic check: a function satisfying Cauchy–Riemann at a point but not differentiable in a neighbourhood is not holomorphic. Stability in control theory: poles on the imaginary axis give marginal stability; require additional analysis. --- 📍 When to Use Which Addition/Subtraction: stay in Cartesian form. Multiplication/Division/Power/Root: switch to polar/exponential form. Finding modulus or conjugate: Cartesian is fastest. Solving for \(\log z\) or \(z^{a}\) with non‑integer exponent: use exponential form \(z = re^{i\theta}\). Testing analyticity: compute partials of \(u, v\) and apply Cauchy–Riemann. Number‑theoretic problems (sums of squares): use Gaussian integer norm and its multiplicative property. --- 👀 Patterns to Recognize \(i\) power pattern: 1, \(i\), \(-1\), \(-i\), then repeat every 4. Modulus‑square identity: \(|z|^{2}=z\overline{z}\). Argument addition: when multiplying, \(\arg(z{1}z{2}) = \arg(z{1})+\arg(z{2})\) (mod \(2\pi\)). Root angles equally spaced: roots of order \(n\) are spaced by \(\frac{2\pi}{n}\). Norm multiplicativity: \(N(z{1}z{2}) = N(z{1})N(z{2})\) → product of two sums of squares is a sum of squares. Cauchy–Riemann symmetry: \(u{x}=v{y}\) and \(u{y}=-v{x}\) appear as a “rotated” gradient pair. --- 🗂️ Exam Traps Wrong sign in conjugate: picking \(\overline{z}=a+bi\) instead of \(a-bi\). Argument off by \(π\): using \(\operatorname{atan}(b/a)\) without quadrant correction; the correct function is \(\operatorname{atan2}(b,a)\). Missing \(2k\pi\) in logarithm: choosing \(\log w = \ln r + i\theta\) and ignoring other branches. Applying De Moivre to non‑integer exponents: the formula holds only for integer \(n\). Assuming real roots for all polynomials: forget the algebraic closure of \(\mathbb{C}\). Stability mis‑classification: marking a system stable because all poles have negative real parts, but overlooking a pole at the origin (marginal case). Gaussian integer factorisation: neglecting units (\(\pm1, \pm i\)) leading to incorrect uniqueness statements. ---