Introductory Riemann surfaces and branched mapsComplex Analysis II · Mixed review
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Showing 15 of 15 matching problems.
Zeros and poles both vote: F=((z-1/2)^2(z+1)(z-3))/(z^2(z-2)) has winding index 1 on |z|=3/2
Factor-mean Jensen for f=2z^3-1 on the unit circle yields mean log 2 and no boundary zeros
Schwarz reflection of z^2+1/(3-z) across (-2,2) continues as the same rational function
The inverse germ of w e^w at 0 has Lagrange coefficients (-n)^{n-1}/n! and branches at -e^{-1} with radius 1/e
On 1<|z|<4 the radial solution u=log|z|/log 4 has flux and Dirichlet energy 2π/log 4
Upper-half-plane Poisson/Cauchy convolution of 1/(1+t^2) is (1+y)/(x^2+(1+y)^2)
For 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)=3
The 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^2/4
The 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 0
The 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}
The 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})]
The 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)|=1
The 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)=1/32
The degree-3 sphere map z^3-3z has ramification 1+1+2=4=2d-2, with simple fiber {0,±sqrt(3)} over 0
On C/(Z+iZ) the form dz descends with periods 1, i, and 1+i, has no global primitive, and induces area 1