https://teddyrookbook.com/math/complex-analysis/Recent content in Complex Analysis on Teddy Rook Bookhttps://teddyrookbook.com/img/share.jpghttps://teddyrookbook.com/img/share.jpgHugoenhttps://teddyrookbook.com/math/complex-analysis/modulus-conjugate/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/modulus-conjugate/|z|² = z·z̄ is a real number, |zw| = |z| |w|, and |z + w| ≤ |z| + |w|. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/exponential-form/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/exponential-form/e^(iθ) = cos θ + i sin θ, so z = re^(iθ); multiplying multiplies the lengths and adds the angles. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/principal-argument/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/principal-argument/arg z is a whole family of angles θ + 2πk; the principal value Arg z lies in (−π, π], and Arg(zw) can differ from Arg z + Arg w by 2π. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/regions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/regions/Read sets such as |z − 2i| < 3 (a disk), 1 < |z| < 2 (an annulus) and Re z > 0 (a half-plane), and tell open, closed, bounded and connected sets apart. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/complex-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/complex-functions/Write f(z) = u(x, y) + i v(x, y): for f(z) = z², u = x² − y² and v = 2xy. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/mappings/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/mappings/Watch f move points from the z-plane to the w-plane: az + b rotates, scales and shifts, z² squares lengths and doubles angles, and 1/z turns the plane inside out. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/complex-limits/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/complex-limits/lim f(z) = L when f(z) approaches L along every path to z₀; z̄/z has no limit at 0. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/complex-derivative/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/complex-derivative/f′(z₀) = lim (f(z₀ + h) − f(z₀))/h as h → 0 from every direction in the plane; z² is differentiable everywhere, and z̄ nowhere. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/cauchy-riemann/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/cauchy-riemann/f = u + iv with continuous partial derivatives is differentiable exactly where uₓ = v_y and u_y = −vₓ, and there f′ = uₓ + i vₓ. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/analytic-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/analytic-functions/f is analytic on an open set when it is differentiable at every point of it; sums, products, quotients and compositions follow the usual rules, and polynomials are entire. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/harmonic-functions/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/harmonic-functions/The real and imaginary parts of an analytic function solve u_xx + u_yy = 0; given u, find v from the Cauchy–Riemann equations. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/complex-exponential/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/complex-exponential/e^z = eˣ(cos y + i sin y): |e^z| = eˣ, e^z is never 0, and it repeats with period 2πi. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/trig-hyperbolic/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/trig-hyperbolic/cos z = (e^(iz) + e^(−iz))/2 and sin z = (e^(iz) − e^(−iz))/(2i) are entire and unbounded, so sin z = 2 has solutions. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/complex-logarithm/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/complex-logarithm/log z = ln|z| + i arg z has infinitely many values, 2πi apart; the principal value is Log z = ln|z| + i Arg z. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/branch-cuts/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/branch-cuts/A branch chooses one value continuously on a cut plane: Log jumps by 2πi across its cut, the negative real axis, and a different cut gives a different branch. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/complex-powers/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/complex-powers/z^c = e^(c log z) can have many values: every value of i^i = e^(−π/2 − 2πk) is a real number. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/contours/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/contours/Describe segments and circles as z(t): z(t) = a + re^(it), 0 ≤ t ≤ 2π, goes once around the circle counterclockwise. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/contour-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/contour-integrals/∫_C f(z) dz = ∫ f(z(t)) z′(t) dt; once around the unit circle, ∮ dz/z = 2πi. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/ml-inequality/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/ml-inequality/|∫_C f(z) dz| ≤ M · L, where M bounds |f| on C and L is the length of C. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/path-independence/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/path-independence/If F′ = f on an open set containing C, then ∫_C f dz = F(end) − F(start); 1/z has no antiderivative on any loop around 0. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/cauchy-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/cauchy-theorem/If f is analytic on and inside a simple closed contour C, then ∮_C f(z) dz = 0. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/deformation/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/deformation/An integral does not change when the contour slides across places where f is analytic, so ∮ dz/(z − a)ⁿ around a is 2πi for n = 1 and 0 for every other integer n. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/cauchy-integral-formula/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/cauchy-integral-formula/If f is analytic on and inside C and a is inside C, then f(a) = (1/(2πi)) ∮_C f(z)/(z − a) dz. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/cauchy-derivatives/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/cauchy-derivatives/∮_C f(z)/(z − a)^(n + 1) dz = 2πi f⁽ⁿ⁾(a)/n!, so an analytic function has derivatives of every order. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/liouville/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/liouville/A bounded entire function is constant, so every nonconstant polynomial has a complex root. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/maximum-modulus/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/maximum-modulus/On a bounded region, a nonconstant analytic function that is continuous up to the boundary reaches its largest |f| on the boundary, never inside. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/complex-power-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/complex-power-series/Σ cₙ(z − a)ⁿ converges inside a disk and diverges outside it; find the radius with the ratio or root test. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/complex-taylor/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/complex-taylor/An analytic function equals its Taylor series on the largest open disk around a that avoids its singularities. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/laurent-series/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/laurent-series/In an annulus, f = Σ cₙ(z − a)ⁿ with negative powers too: 1/(z(z − 1)) has one series on 0 < |z| < 1 and another on |z| > 1. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/singularities/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/singularities/Classify an isolated singularity as removable, a pole of order m or essential, from the Laurent series or from limits. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/residues/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/residues/The residue at a is the coefficient of 1/(z − a); at a simple pole it is lim (z − a) f(z), which is g(a)/h′(a) when f = g/h. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/higher-order-residues/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/higher-order-residues/At a pole of order m, the residue is (1/(m − 1)!) times the limit of the (m − 1)st derivative of (z − a)ᵐ f(z). Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/residue-theorem/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/residue-theorem/∮_C f(z) dz = 2πi times the sum of the residues of f at the poles inside C. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/trig-integrals-residues/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/trig-integrals-residues/Turn ∫₀^{2π} R(cos θ, sin θ) dθ into a contour integral around the unit circle with z = e^(iθ), cos θ = (z + 1/z)/2 and dθ = dz/(iz). Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/real-integrals-residues/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/real-integrals-residues/Close the real line with a large half-circle in the upper half-plane; when deg Q ≥ deg P + 2, the arc adds nothing as the radius grows, so ∫ P/Q dx is 2πi times the upper residues. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/fourier-type-integrals/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/fourier-type-integrals/For ∫ R(x) cos(ax) dx with a > 0, integrate R(z) e^(iaz) around the upper half-disk and take the real part. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/argument-principle/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/argument-principle/(1/(2πi)) ∮_C f′/f dz counts the zeros minus the poles inside C, which is the number of times f(C) winds around 0. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/rouche/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/rouche/If |g| < |f| everywhere on C, then f and f + g have the same number of zeros inside C. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/conformal-maps/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/conformal-maps/Where f is analytic and f′ ≠ 0, f keeps angles: tiny shapes are turned by arg f′ and scaled by |f′|. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/mobius/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/mobius/w = (az + b)/(cz + d) with ad − bc ≠ 0 maps circles and lines to circles and lines, and, when c ≠ 0, sends −d/c to ∞ and ∞ to a/c. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/mobius-three-points/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/mobius-three-points/Exactly one Möbius transformation sends three given points to three given points; find it with the cross-ratio. Complex Analysis · Math Ladderhttps://teddyrookbook.com/math/complex-analysis/standard-maps/Mon, 01 Jan 0001 00:00:00 +0000https://teddyrookbook.com/math/complex-analysis/standard-maps/Send the upper half-plane to the unit disk with (z − i)/(z + i), open sectors with zⁿ and horizontal strips with e^z, and chain maps together. Complex Analysis · Math Ladder