Holomorphic function Study Guide

📖 Core Concepts Holomorphic function – complex‑valued function differentiable in a neighbourhood of every point of its domain. Complex differentiability at \(z0\) – the limit \[ \lim{z\to z0}\frac{f(z)-f(z0)}{z-z0} \] exists and is independent of the direction of approach. Analytic vs. Holomorphic – In complex analysis the two are equivalent: a function is holomorphic iff it can be expressed locally by a convergent power series (its Taylor series). Cauchy–Riemann (CR) equations – For \(f=u+iv\), \[ \frac{\partial u}{\partial x}= \frac{\partial v}{\partial y},\qquad \frac{\partial u}{\partial y}= -\frac{\partial v}{\partial x}. \] If \(u,v\) have continuous partials and satisfy these, \(f\) is holomorphic. Entire function – holomorphic on the whole complex plane \(\mathbb{C}\). Harmonic real/imaginary parts – If \(f\) is holomorphic, both \(u\) and \(v\) satisfy Laplace’s equation \(\Delta u=\Delta v=0\). Conformal mapping – Where \(f'(z)\neq0\), a holomorphic map preserves angles locally. --- 📌 Must Remember Holomorphic ⇔ analytic (complex case). CR equations are necessary (and with continuity, sufficient) for holomorphy. Cauchy’s Integral Theorem: \(\displaystyle\oint{\gamma}f(z)\,dz=0\) for any closed path \(\gamma\) in a simply‑connected holomorphic domain. Cauchy’s Integral Formula: \(\displaystyle f(z)=\frac{1}{2\pi i}\oint{\partial D}\frac{f(\zeta)}{\zeta-z}\,d\zeta\). Derivative formula: \(\displaystyle f^{(n)}(z)=\frac{n!}{2\pi i}\oint{\partial D}\frac{f(\zeta)}{(\zeta-z)^{n+1}}\,d\zeta\). Algebraic closure: sums, products, compositions, and quotients (where denominator \(\neq0\)) of holomorphic functions remain holomorphic. Entire examples: polynomials, \(e^{z}\), \(\sin z\), \(\cos z\). Non‑holomorphic facts: \(|z|\), \(\operatorname{Re}z\), \(\operatorname{Im}z\), \(\arg z\) are not holomorphic; a real‑valued holomorphic function must be constant. --- 🔄 Key Processes Checking holomorphy via CR: Compute \(\partial u/\partial x,\ \partial u/\partial y,\ \partial v/\partial x,\ \partial v/\partial y\). Verify the two CR equalities; confirm continuity of the partials. Using Cauchy’s Integral Formula: Identify a closed disk \(D\) containing the point \(z\) with \(\partial D\) inside the holomorphic region. Evaluate \(\displaystyle \frac{1}{2\pi i}\oint{\partial D}\frac{f(\zeta)}{\zeta-z}\,d\zeta\). Finding higher derivatives: Apply the derivative formula with the same contour: replace \((\zeta-z)\) by \((\zeta-z)^{n+1}\) and multiply by \(n!\). Constructing a holomorphic function from a harmonic function: Given harmonic \(u\) on a simply connected domain, find a harmonic conjugate \(v\) (solve CR for \(v\)), then \(f=u+iv\) is holomorphic, unique up to an additive constant. --- 🔍 Key Comparisons Holomorphic vs. Analytic – A = complex‑differentiable locally; B = representable by a convergent power series. In \(\mathbb{C}\) they are equivalent. Entire vs. Holomorphic – Entire = holomorphic on all of \(\mathbb{C}\); Holomorphic may have a smaller domain. Harmonic vs. Holomorphic parts – Harmonic: satisfies \(\Delta u=0\); Holomorphic: both real and imaginary parts are harmonic and linked by CR. Holomorphic vs. Antiholomorphic – Holomorphic: \(\partial f/\partial\overline{z}=0\); Antiholomorphic: \(\partial f/\partial z=0\) (e.g., \(\overline{z}\)). --- ⚠️ Common Misunderstandings “Differentiable at a point ⇒ holomorphic.” False: a function can have a complex derivative at a single point without being differentiable in a neighbourhood. “All real‑valued holomorphic functions are non‑constant.” Wrong; they must be constant by the Cauchy–Riemann equations. “\(|z|\) is holomorphic because it’s smooth.” Incorrect; it fails the CR equations. Confusing domain of \(\log z\) with whole \(\mathbb{C}\). The principal branch is holomorphic only on \(\mathbb{C}\setminus(-\infty,0]\). --- 🧠 Mental Models / Intuition Holomorphic = “smooth in every complex direction.” Imagine probing the function with tiny arrows from every angle; the limit of the difference quotient must be the same. CR equations = “rotation of the gradient.” The gradient of \(u\) is a 90° rotation of the gradient of \(v\). Cauchy’s theorem = “no net circulation.” In a simply connected holomorphic region, the integral around any closed loop cancels out—like a frictionless fluid. Conformal = “angle‑preserving microscope.” Near any point where \(f'(z)\neq0\), the map looks like a rotation + scaling, never shearing. --- 🚩 Exceptions & Edge Cases Zero derivative: If \(f'(z)=0\) at a point, the map is not conformal there (angles are preserved but local scaling collapses). Branch cuts: Functions like \(\log z\) and \(\sqrt{z}\) are holomorphic only after removing a branch cut (e.g., \((-∞,0]\)). Simply connected requirement: Cauchy’s theorem and the existence of a harmonic conjugate both need a simply connected domain. Meromorphic vs. holomorphic: Rational functions are holomorphic except at poles; they become meromorphic on the whole plane. --- 📍 When to Use Which CR test – When you have an explicit \(u(x,y),v(x,y)\) and need a quick holomorphy check. Cauchy’s Integral Theorem – To show an integral over a closed path is zero or to deform contours. Cauchy’s Integral Formula – To evaluate \(f(z)\) or its derivatives inside a known contour. Algebraic closure – When combining known holomorphic functions (sum, product, composition) you can immediately claim the result is holomorphic. Harmonic conjugate method – When given a harmonic real part and asked to produce a holomorphic function. --- 👀 Patterns to Recognize Zero‑integral loops → the integrand is holomorphic on the interior (Cauchy’s theorem). Denominator \((\zeta-z)^{n+1}\) in a contour integral → you are looking at the \(n^{\text{th}}\) derivative (Cauchy’s differentiation formula). Presence of \(\partial f/\partial\overline{z}\) – if it vanishes, the function is holomorphic. Polynomials, exponentials, trigonometric functions → automatically entire; no need to check CR. --- 🗂️ Exam Traps Choosing “analytic” instead of “holomorphic.” On a complex‑analysis exam they are interchangeable, but the wording may be testing your awareness of the theorem. Assuming \(|z|\) is holomorphic because it’s smooth – it fails CR; answer choice will be wrong. For \(\log z\) forgetting the branch cut – an answer that claims \(\log z\) is holomorphic on all of \(\mathbb{C}\) is a trap. Using Cauchy’s Integral Formula on a non‑simply‑connected domain – the formula requires a contour that encloses a region where the function is holomorphic. Treating a real‑valued holomorphic function as possibly non‑constant – the only correct choice is “constant.” ---