CA:020117Zeros and poles both vote: F=((z-1/2)^2(z+1)(z-3))/(z^2(z-2)) has winding index 1 on |z|=3/2Complex Analysis 2 Argument Principle WindingComplex Analysis 2 Logarithmic Derivative ResiduesPrivateLabels unavailablePrivate saving unavailableArgument principle, Rouché, and zero countingComplex Analysis II · Mixed reviewProgress not loaded
CA:020118Factor-mean Jensen for f=2z^3-1 on the unit circle yields mean log 2 and no boundary zerosComplex Analysis 2 Jensen Formula Unit DiskComplex Analysis 2 Mean Log FactorizationPrivateLabels unavailablePrivate saving unavailableArgument principle, Rouché, and zero countingComplex Analysis II · Mixed reviewProgress not loaded
CA:020219Schwarz reflection of z^2+1/(3-z) across (-2,2) continues as the same rational functionComplex Analysis 2 Schwarz Reflection PrincipleComplex Analysis 2 Rational Analytic ContinuationPrivateLabels unavailablePrivate saving unavailableAnalytic continuation and monodromyComplex Analysis II · Mixed reviewProgress not loaded
CA:020228The inverse germ of w e^w at 0 has Lagrange coefficients (-n)^{n-1}/n! and branches at -e^{-1} with radius 1/eComplex Analysis 2 Lagrange Inversion LambertComplex Analysis 2 Puiseux Branch PointPrivateLabels unavailablePrivate saving unavailableAnalytic continuation and monodromyComplex Analysis II · Mixed reviewProgress not loaded
CA:020320On 1<|z|<4 the radial solution u=log|z|/log 4 has flux and Dirichlet energy 2π/log 4Complex Analysis 2 Annulus Dirichlet RadialComplex Analysis 2 Harmonic Flux EnergyPrivateLabels unavailablePrivate saving unavailableHarmonic functions and boundary-value methodsComplex Analysis II · Mixed reviewProgress not loaded
CA:020321Upper-half-plane Poisson/Cauchy convolution of 1/(1+t^2) is (1+y)/(x^2+(1+y)^2)Complex Analysis 2 Uhp Poisson IntegralComplex Analysis 2 Cauchy Kernel ConvolutionPrivateLabels unavailablePrivate saving unavailableHarmonic functions and boundary-value methodsComplex Analysis II · Mixed reviewProgress not loaded
CA:020327For a=1/2 the disk Green function vanishes on the circle, is positive inside, and its negative outward normal derivative is the Poisson kernel with P_a(1)=3Complex Analysis 2 Disk Green Boundary PositivityComplex Analysis 2 Green Normal Poisson KernelPrivateLabels unavailablePrivate saving unavailableHarmonic functions and boundary-value methodsComplex Analysis II · Mixed reviewProgress not loaded
CA:020422The single-valued series F(z)=sum (-1)^n z^n/(2n)! is entire of order 1/2 and type 1, with F(0)=1, F'(0)=-1/2, and first positive zeros pi^2/4, 9pi^2/4, 25pi^2/4Complex Analysis 2 Entire Series Order TypeComplex Analysis 2 Cos Sqrt Simple ZerosPrivateLabels unavailablePrivate saving unavailableEntire and meromorphic function structureComplex Analysis II · Mixed reviewProgress not loaded
CA:020423The rational R=z^2/((z-1)(z+2)(z-3)) has residues -1/6, 4/15, 9/10 at 1, -2, 3, finite sum 1, residue at infinity -1, and sphere total 0Complex Analysis 2 Finite Residues Two WaysComplex Analysis 2 Residue AT Infinity SpherePrivateLabels unavailablePrivate saving unavailableEntire and meromorphic function structureComplex Analysis II · Mixed reviewProgress not loaded
CA:020525The Laplace integral I_n=int_0^1 e^{-n x}(1+x) dx equals 1/n+1/n^2-e^{-n}(2/n+1/n^2), with n I_n->1, n^2(I_n-1/n)->1, and I_2=3/4-(5/4)e^{-2}Complex Analysis 2 Watson Endpoint ExpansionComplex Analysis 2 Exponentially Small RemainderPrivateLabels unavailablePrivate saving unavailableAsymptotic contour methodsComplex Analysis II · Mixed reviewProgress not loaded
CA:020526The generating function (1-z)^{-3/2} has coefficients a_n=Gamma(n+3/2)/(Gamma(3/2) n!)=(2n+1)C(2n,n)/4^n, with a0..a3=(1, 3/2, 15/8, 35/16), ratio (n+3/2)/(n+1), and Darboux expansion (2/sqrt(pi)) n^{1/2}[1+3/(8n)-7/(128 n^2)+O(n^{-3})]View access optionsComplex Analysis 2 Generalized Binomial CoefficientsComplex Analysis 2 Darboux Stirling AsymptoticsPrivateLabels unavailablePrivate saving unavailableAsymptotic contour methodsComplex Analysis II · Mixed reviewProgress not loaded
CA:020530The endpoint integral of x e^{i lambda x} on [0,1] is exact of size O(lambda^{-1}) with no saddle, and |lambda J(2 pi n)|=1View access optionsComplex Analysis 2 Endpoint Integration BY PartsComplex Analysis 2 Nonstationary Phase BoundPrivateLabels unavailablePrivate saving unavailableAsymptotic contour methodsComplex Analysis II · Mixed reviewProgress not loaded
CA:020624The rational h=(z-1)^2/(z^3(z+2)) has orders +2, -3, -1, +2 at 1, 0, -2, infinity, divisor degree 0, total zero/pole degree 4, and h(2)=1/32View access optionsComplex Analysis 2 Rational Local OrdersComplex Analysis 2 Divisor Degree SpherePrivateLabels unavailablePrivate saving unavailableIntroductory Riemann surfaces and branched mapsComplex Analysis II · Mixed reviewProgress not loaded
CA:020629The degree-3 sphere map z^3-3z has ramification 1+1+2=4=2d-2, with simple fiber {0,±sqrt(3)} over 0View access optionsComplex Analysis 2 Local Degree Critical ValuesComplex Analysis 2 Riemann Hurwitz SpherePrivateLabels unavailablePrivate saving unavailableIntroductory Riemann surfaces and branched mapsComplex Analysis II · Mixed reviewProgress not loaded
CA:020631On C/(Z+iZ) the form dz descends with periods 1, i, and 1+i, has no global primitive, and induces area 1View access optionsComplex Analysis 2 Lattice Invariant FormComplex Analysis 2 Period Lattice ExactnessPrivateLabels unavailablePrivate saving unavailableIntroductory Riemann surfaces and branched mapsComplex Analysis II · Mixed reviewProgress not loaded