Mathematical Methods II — Complex Analysis & Asymptotics — formula sheet
Every boxed result from the 18 derivations of this unit on one page, for revision. Each title links to the full derivation, with its assumptions and where the formula breaks down. Constants: physical constants table.
Cauchy-Riemann Equations & Harmonic Conjugates
Complex differentiability is not a mild smoothness condition — it is a system of coupled first-order partial differential equations. The real and imaginary parts of an analytic function cannot be chosen independently: each determines the other up to an additive constant (on a simply connected domain), each solves Laplace's equation, and their level sets form two mutually orthogonal families of curves. The derivative itself can be read off either family: \( f'(z) = u_x + i v_x = v_y - i u_y \).
Cauchy-Goursat Theorem
An analytic function has no "circulation" and no "flux" around any closed loop inside its domain of analyticity: closed-loop integrals detect only obstructions — singularities or holes in the domain — and here there are none. Equivalently, contour integrals of analytic functions are path-independent, analytic functions always admit local antiderivatives, and integration contours may be freely deformed without changing the value of the integral, provided the deformation never crosses a singularity or leaves the domain.
Cauchy Integral Formula & Derivative Formula
The value of an analytic function — and every one of its derivatives — at any interior point is a weighted average of its values on the surrounding contour, with weight \( n!/\bigl(2\pi i\,(z-z_0)^{n+1}\bigr) \). The case \( n=0 \) is the Cauchy integral formula; the interior is a hologram of the boundary. Immediate consequences: analytic once means analytic infinitely often; bounded entire functions are constant (Liouville); every nonconstant polynomial has a complex root.
Taylor Series of Analytic Functions
A holomorphic function is locally identical to its Taylor series, not merely approximated by it. The single hypothesis of complex differentiability on an open set upgrades to full analyticity, and the reach of the expansion is dictated globally by where the function first misbehaves — the nearest pole, branch point, or essential singularity — regardless of how well-behaved \(f\) looks near \(z_0\).
Laurent Series & Classification of Singularities
Any function holomorphic on an annulus is exactly a sum of a Taylor-like part (non-negative powers, convergent inside \(R_2\)) and a principal part (negative powers, convergent outside \(R_1\)). The coefficients are unique and are contour integrals of \(f\). The lone coefficient \(a_{-1}\) is the residue. The tail of the principal part is a complete diagnostic: empty \(\Rightarrow\) the singularity was an illusion (removable); finite of length \(m\) \(\Rightarrow\) pole of order \(m\); infinite \(\Rightarrow\) essential, where \(f\) behaves as wildly as any holomorphic function can.
The Residue Theorem
The whole contour integral depends only on the sum of residues — the \(a_{-1}\) Laurent coefficients — of the singularities the curve encloses. Regular parts and all other Laurent powers contribute nothing; the loop "counts" singularities weighted by their \((z-z_k)^{-1}\) strength, each worth exactly \(2\pi i\) times its residue.
Real Integrals by Residues & Jordan's Lemma
An oscillatory integral over the entire real line equals \(2\pi i\) times the sum of residues at the poles of \(f\) lying strictly above the axis — provided the oscillation phase \(a>0\) lets us close upward, where \(e^{iaz}\) decays exponentially. Jordan's lemma is precisely the guarantee that the "imaginary" arc we appended contributes nothing, so the answer is entirely local data (residues) even though the integral is global.
Argument Principle & Rouché's Theorem
The contour integral of the logarithmic derivative of \(f\) equals the net number of zeros minus poles trapped inside the loop — equivalently, the number of times the image curve \(f(C)\) circles the origin. Rouché says that if a "perturbation" \(g-f\) is uniformly smaller in magnitude than \(f\) itself all along the boundary, the perturbation cannot create or destroy any enclosed root, so the simpler comparison function \(f\) already tells you the count.
Branch Points, Cuts & Riemann Surfaces
The logarithm's value is fixed up to integer multiples of \( 2\pi i \); which multiple you hold depends on how many times your path has wound around the branch point \( z=0 \). Cutting the plane from \( 0 \) to \( \infty \) makes a single branch well defined; stacking the branches across the cut builds the Riemann surface on which the function is a single-valued holomorphic map. A power \( z^{p/q} \) closes up after \( q \) turns, giving a \( q \)-sheeted surface.
Analytic Continuation & the Identity Theorem
Two analytic functions that coincide on any set thick enough to accumulate at an interior point are the same function everywhere their common domain reaches. In particular the germ at one point — the value and all derivatives, equivalently one Taylor series — determines \(f\) globally, and analytic continuation is unique whenever the target domain is connected.
Gamma Function: Continuation & Reflection Formula
The product of the Gamma function at a point and at its reflection through \( z=\tfrac12 \) is a pure trigonometric object. The poles of \( 1/\sin(\pi z) \) at every integer are supplied on the left by \( \Gamma(z) \) at \( z=0,-1,-2,\dots \) and by \( \Gamma(1-z) \) at \( z=1,2,3,\dots \), so each integer pole comes from exactly one factor. Because \( \pi/\sin(\pi z) \) never vanishes, neither \( \Gamma(z) \) nor \( \Gamma(1-z) \) has any zero — the Gamma function is zero-free on all of \( \mathbb{C} \).
Conformal Mapping & Plane Potential Theory
An analytic map with non-vanishing derivative rotates-and-scales every infinitesimal neighbourhood by the single complex number \(f'(z)\), so it preserves angles (conformal). The Laplacian merely picks up the positive real factor \(|f'(z)|^2\); therefore a harmonic potential stays harmonic under the map. To solve \(\Delta\Phi=0\) on an awkward domain \(D\), map \(D\) conformally to a simple domain, carry the boundary values along, solve the easy Laplace problem there, and pull the solution back.
Asymptotic Series & Optimal Truncation
Stop a divergent asymptotic series at its smallest term, near order \(N^*\approx|A|x\). The error incurred is about the size of that smallest term. For factorial coefficients this least-term error is exponentially small in \(x\), so a divergent series gives super-algebraic accuracy — but no better, because the exponentially small piece \(e^{-|A|x}\) is invisible to the power series and sets a hard floor.
Watson's Lemma
Each power \(t^{\alpha_n}\) in the integrand's endpoint expansion is replaced by its own Laplace transform \(\Gamma(\alpha_n+1)\,\lambda^{-(\alpha_n+1)}\), and you simply add them up. The leading term \(a_0\,\Gamma(\alpha_0+1)\,\lambda^{-(\alpha_0+1)}\) fixes both the rate of decay of \(I(\lambda)\) and its amplitude; higher terms are successive corrections in inverse powers of \(\lambda\). Only the behaviour of \(f\) at \(t=0\) matters — the rest of the profile is exponentially invisible.
Laplace's Method for Exponential Integrals
The whole integral is set by the single dominant point \(t_0\): an exponential factor \(e^{\lambda\varphi(t_0)}\) fixing the order of magnitude, the value \(g(t_0)\) of the slowly-varying weight there, and a Gaussian "volume of fluctuations" \(\sqrt{2\pi/\lambda\lvert\varphi''\rvert}\) whose width \(\sim(\lambda\lvert\varphi''\rvert)^{-1/2}\) shrinks as \(\lambda\) grows or as the peak sharpens. Flatter peaks (small \(\lvert\varphi''\rvert\)) contribute more; sharper peaks contribute less.
Method of Steepest Descent
The answer factorises into three transparent pieces: the amplitude of the smooth part at the saddle, \( g(z_0) \); the exponential set by the saddle height, \( e^{\lambda f(z_0)} \); and a Gaussian width factor \( \sqrt{2\pi/(\lambda|f''|)} \) that shrinks like \( \lambda^{-1/2} \) because the peak grows sharper with \( \lambda \). The phase \( e^{i\theta} \) records the tilt of the steepest-descent line; it is the term Laplace's real-axis method never sees, and it is exactly the \( \pi/4 \) Fresnel phase in diffraction. Equivalently one writes \( I\sim g(z_0)\,e^{\lambda f(z_0)}\sqrt{-2\pi/\big(\lambda f''(z_0)\big)} \) with the branch of the root fixed by \( \theta \).
Stirling's Formula via Saddle Point
In statistical mechanics only the logarithm is needed, and only the first two terms of it: the √(2πN) contributes ½ ln(2πN), which is of order ln N against terms of order N and is discarded without hesitation. But it must be derived before it can be discarded honestly.
Method of Stationary Phase
The whole integral is governed by the single point where the phase stops changing. The amplitude there, \(g(t_0)\), is weighted by a "resonance width" \(\sqrt{2\pi/(\lambda|\phi''|)}\): the flatter the phase (small \(\phi''\)) or the slower the oscillation (small \(\lambda\)), the wider the constructively-adding window and the larger the result. The output oscillates at the stationary phase value \(\lambda\phi(t_0)\), and — crucially — carries an extra fixed \(\pm45^\circ\) phase kick set only by whether \(t_0\) is a phase minimum (\(+\)) or maximum (\(-\)).