Spin-weighted spherical harmonics
This example plays with analysis of:
\[f(r) = e^{{\rm i} k\cdot r},\]
for some $k\in\mathbb{R}^3$ and where $r\in\mathbb{S}^2$, using spin-$0$ spherical harmonics.
It applies ð, the spin-raising operator, both on the spin-$0$ coefficients as well as the original function, followed by a spin-$1$ analysis to compare coefficients.
For the storage pattern of the arrays, please consult the documentation.
using FastTransforms, LinearAlgebraThe colatitudinal grid (mod $\pi$):
N = 10
θ = (0.5:N-0.5)/N0.05:0.1:0.95The longitudinal grid (mod $\pi$):
M = 2*N-1
φ = (0:M-1)*2/M0.0:0.10526315789473684:1.894736842105263Our choice of $k$ and angular parametrization of $r$:
k = [2/7, 3/7, 6/7]
r = (θ,φ) -> [sinpi(θ)*cospi(φ), sinpi(θ)*sinpi(φ), cospi(θ)]#1 (generic function with 1 method)On the tensor product grid, our function samples are:
F = [exp(im*(k⋅r(θ,φ))) for θ in θ, φ in φ]10×19 Matrix{ComplexF64}:
0.628413+0.77788im 0.613246+0.789892im 0.603399+0.79744im 0.600106+0.79992im 0.603786+0.797146im 0.613972+0.789328im 0.629387+0.777092im 0.64815+0.761513im 0.668065+0.744103im 0.686931+0.726723im 0.702801+0.711386im 0.71416+0.699982im 0.720001+0.693973im 0.71983+0.694151im 0.713662+0.700491im 0.702017+0.71216im 0.685934+0.727664im 0.666955+0.745098im 0.647044+0.762452im
0.626742+0.779227im 0.582025+0.813171im 0.552374+0.833596im 0.542362+0.840145im 0.55355+0.832816im 0.584193+0.811615im 0.629571+0.776943im 0.68294+0.730474im 0.736952+0.675945im 0.785185+0.619262im 0.823211+0.567736im 0.848825+0.528674im 0.861428+0.507879im 0.861065+0.508495im 0.84773+0.530428im 0.821392+0.570364im 0.782715+0.62238im 0.734019+0.679129im 0.679858+0.733344im
0.690857+0.722991im 0.625071+0.780568im 0.580335+0.814378im 0.565059+0.82505im 0.582122+0.813101im 0.628309+0.777964im 0.694941+0.719066im 0.769832+0.638246im 0.840471+0.541857im 0.897514+0.440985im 0.936972+0.349406im 0.959893+0.280368im 0.969821+0.243819im 0.969549+0.244899im 0.958985+0.283457im 0.935221+0.354065im 0.894765+0.446538im 0.836798+0.547511im 0.76563+0.643282im
0.799876+0.600166im 0.729022+0.68449im 0.678871+0.734257im 0.661444+0.749994im 0.680901+0.732375im 0.732596+0.680663im 0.804139+0.594441im 0.878702+0.477371im 0.940401+0.340068im 0.97989+0.199537im 0.997175+0.0751106im 0.999864-0.0165003im 0.997938-0.0641824im 0.998027-0.062782im 0.999923-0.0124432im 0.996684+0.0813636im 0.978303+0.207181im 0.937478+0.348046im 0.874727+0.484617im
0.914598+0.404364im 0.858511+0.512795im 0.815749+0.578405im 0.800419+0.599441im 0.81752+0.5759im 0.861473+0.507803im 0.917769+0.397116im 0.967817+0.251653im 0.996186+0.0872572im 0.99722-0.07452im 0.977203-0.212305im 0.950621-0.310353im 0.93288-0.360187im 0.933439-0.358735im 0.952008-0.306074im 0.978656-0.205506im 0.997827-0.065887im 0.99532+0.0966353im 0.965471+0.260509im
0.989052+0.147569im 0.963701+0.266985im 0.939852+0.341582im 0.930644+0.365927im 0.940896+0.338696im 0.965234+0.261387im 0.990188+0.13974im 0.999905-0.0137846im 0.983699-0.179822im 0.94183-0.33609im 0.88602-0.463647im 0.834409-0.551146im 0.804097-0.594498im 0.805021-0.593247im 0.836879-0.547388im 0.889222-0.457475im 0.944703-0.327927im 0.985349-0.170549im 0.999989-0.00462386im
0.99096-0.134156im 0.999703-0.0243628im 0.998928+0.0462861im 0.997566+0.0697274im 0.999053+0.0435207im 0.999562-0.0295972im 0.989978-0.141225im 0.960887-0.276941im 0.908431-0.418034im 0.837893-0.545834im 0.762849-0.646576im 0.700449-0.713703im 0.665602-0.746307im 0.666649-0.745372im 0.703339-0.710855im 0.76689-0.641779im 0.842129-0.539276im 0.91195-0.410301im 0.963143-0.26899im
0.919472-0.393156im 0.950297-0.311346im 0.966254-0.257589im 0.970887-0.23954im 0.965686-0.259711im 0.948995-0.315292im 0.91723-0.398357im 0.867837-0.49685im 0.802371-0.596825im 0.727938-0.685643im 0.656033-0.754733im 0.599431-0.800426im 0.568692-0.82255im 0.569608-0.821916im 0.602006-0.798492im 0.659781-0.751458im 0.732172-0.68112im 0.806377-0.591401im 0.871077-0.491147im
0.80566-0.592379im 0.837633-0.546233im 0.856754-0.515725im 0.862863-0.505438im 0.856026-0.516933im 0.836173-0.548466im 0.8035-0.595305im 0.759389-0.650637im 0.707251-0.706962im 0.652712-0.757606im 0.602883-0.79783im 0.564968-0.825113im 0.544739-0.838606im 0.545338-0.838216im 0.566673-0.823943im 0.605429-0.795899im 0.655721-0.755004im 0.710305-0.703894im 0.762122-0.647434im
0.695346-0.718675im 0.70912-0.705088im 0.717813-0.696236im 0.720677-0.693271im 0.717475-0.696585im 0.708471-0.70574im 0.694445-0.719545im 0.676692-0.736266im 0.656981-0.753907im 0.637428-0.77051im 0.620269-0.784389im 0.607563-0.794272im 0.600886-0.799335im 0.601083-0.799186im 0.608129-0.793839im 0.621132-0.783706im 0.638483-0.769636im 0.658105-0.752927im 0.67776-0.735284imWe precompute a spin-$0$ spherical harmonic–Fourier plan:
P = plan_spinsph2fourier(F, 0)FastTransforms Spin-weighted spherical harmonic--Fourier plan for 10×19-element array of ComplexF64And an FFTW Fourier analysis plan on $\mathbb{S}^2$:
PA = plan_spinsph_analysis(F, 0)FastTransforms plan for FFTW Fourier analysis on the sphere (spin-weighted) for 10×19-element array of ComplexF64Its spin-$0$ spherical harmonic coefficients are:
U⁰ = P\(PA*F)10×19 Matrix{ComplexF64}:
2.98294-8.51991e-17im -0.560381+0.373587im 0.560381+0.373587im 0.0307268-0.0737442im 0.0307268+0.0737442im 0.00123905+0.0063329im -0.00123905+0.0063329im -0.000278656-0.000280997im -0.000278656+0.000280997im 1.91763e-5+3.91877e-6im -1.91763e-5+3.91877e-6im -7.51512e-7+3.05775e-7im -7.51512e-7-3.05775e-7im 1.62607e-8-2.39542e-8im -1.62607e-8-2.39542e-8im -7.57573e-12+9.06989e-10im -7.6043e-12-9.06983e-10im -1.32856e-11-2.14392e-11im 1.44574e-11-2.09394e-11im
-9.83939e-17+1.585im -0.147488-0.221233im -0.147488+0.221233im 0.0242802+0.0101168im -0.0242802+0.0101168im -0.001828+0.000357651im -0.001828-0.000357651im 7.31346e-5-7.25251e-5im -7.31346e-5-7.25251e-5im -9.36425e-7+4.58234e-6im -9.36425e-7-4.58234e-6im -6.79393e-8-1.66977e-7im 6.79393e-8-1.66977e-7im 4.99533e-9+3.39094e-9im 4.99533e-9-3.39094e-9im -1.78808e-10-1.4939e-12im 1.78807e-10-1.49874e-12im 3.97537e-12-2.47069e-12im 3.88749e-12+2.67669e-12im
-0.296042+1.9373e-16im 0.0419095-0.0279397im -0.0419095-0.0279397im -0.00179667+0.00431201im -0.00179667-0.00431201im -5.92556e-5-0.000302862im 5.92556e-5-0.000302862im 1.12623e-5+1.13569e-5im 1.12623e-5-1.13569e-5im -6.70874e-7-1.37097e-7im 6.70874e-7-1.37097e-7im 2.31726e-8-9.42846e-9im 2.31726e-8+9.42846e-9im -4.48157e-10+6.60198e-10im 4.48157e-10+6.60199e-10im 1.83296e-13-2.21049e-11im 1.86671e-13+2.21043e-11im 2.77146e-13+4.85403e-13im -3.40624e-13+4.58315e-13im
7.69323e-17-0.0243812im 0.00315419+0.00473128im 0.00315419-0.00473128im -0.000498246-0.000207603im 0.000498246-0.000207603im 3.42215e-5-6.6955e-6im 3.42215e-5+6.6955e-6im -1.23971e-6+1.22938e-6im 1.23971e-6+1.22938e-6im 1.44141e-8-7.05344e-8im 1.44141e-8+7.05344e-8im 9.54963e-10+2.34704e-9im -9.54963e-10+2.34704e-9im -6.22737e-11-4.22728e-11im -6.22737e-11+4.22728e-11im 2.09765e-12+1.71393e-14im -2.09747e-12+1.79766e-14im -5.56471e-13+3.3894e-13im -5.39577e-13-3.78603e-13im
-0.000199834-6.42779e-17im -0.000327232+0.000218154im 0.000327232+0.000218154im 1.67027e-5-4.00864e-5im 1.67027e-5+4.00864e-5im 5.5389e-7+2.83099e-6im -5.5389e-7+2.83099e-6im -1.01124e-7-1.01974e-7im -1.01124e-7+1.01974e-7im 5.69484e-9+1.16377e-9im -5.69484e-9+1.16377e-9im -1.69612e-10+6.90117e-11im -1.69612e-10-6.90117e-11im 2.98197e-12-4.39287e-12im -2.98201e-12-4.39288e-12im -3.48909e-14+4.21983e-12im -3.57157e-14-4.21966e-12im -4.58893e-14-8.03264e-14im 5.6381e-14-7.5835e-14im
-6.51867e-17-0.000282488im -7.0029e-6-1.05044e-5im -7.0029e-6+1.05044e-5im 2.24809e-6+9.36704e-7im -2.24809e-6+9.36704e-7im -1.76539e-7+3.45402e-8im -1.76539e-7-3.45402e-8im 6.55393e-9-6.49931e-9im -6.55393e-9-6.49931e-9im -7.13679e-11+3.49235e-10im -7.1368e-11-3.49235e-10im -4.99715e-12-1.22817e-11im 4.99718e-12-1.22817e-11im 1.69023e-11+1.14737e-11im 1.69023e-11-1.14737e-11im -5.44917e-13-4.41379e-15im 5.44856e-13-4.70669e-15im 1.5798e-13-9.50585e-14im 1.52381e-13+1.08137e-13im
3.58969e-5-4.03624e-17im -4.7402e-7+3.16014e-7im 4.7402e-7+3.16014e-7im -3.23608e-8+7.76658e-8im -3.23608e-8-7.76658e-8im -1.6111e-9-8.2345e-9im 1.6111e-9-8.2345e-9im 2.28327e-10+2.30246e-10im 2.28327e-10-2.30246e-10im -1.02557e-11-2.09579e-12im 1.02557e-11-2.09576e-12im 4.54949e-11-1.8511e-11im 4.54949e-11+1.85109e-11im -5.65258e-13+8.32684e-13im 5.65261e-13+8.3268e-13im 9.40322e-15-1.14115e-12im 9.65269e-15+1.14112e-12im 8.76404e-15+1.53587e-14im -1.07761e-14+1.4515e-14im
1.02759e-16+2.64492e-6im -5.98533e-8-8.978e-8im -5.98533e-8+8.978e-8im -2.2753e-10-9.48042e-11im 2.2753e-10-9.48042e-11im 2.65455e-10-5.19369e-11im 2.65455e-10+5.19368e-11im -1.37076e-11+1.35933e-11im 1.37076e-11+1.35933e-11im 3.35969e-11-1.64405e-10im 3.35969e-11+1.64405e-10im 1.96453e-12+4.82831e-12im -1.96456e-12+4.82829e-12im -8.01172e-12-5.43855e-12im -8.01172e-12+5.43851e-12im 2.15355e-13+1.72018e-15im -2.15344e-13+1.85439e-15im -7.69555e-14+4.59648e-14im -7.40025e-14-5.28722e-14im
-1.25204e-7-1.00025e-16im 6.65757e-9-4.43838e-9im -6.65757e-9-4.43838e-9im -1.43603e-11+3.44646e-11im -1.43603e-11-3.44647e-11im 5.78456e-14+2.95764e-13im -5.78679e-14+2.9576e-13im -3.63314e-11-3.66368e-11im -3.63314e-11+3.66367e-11im 7.38143e-13+1.50851e-13im -7.3814e-13+1.50867e-13im -7.24159e-12+2.94646e-12im -7.24159e-12-2.94644e-12im 4.07073e-14-5.99537e-14im -4.06954e-14-5.99558e-14im -1.50601e-15+1.8179e-13im -1.53602e-15-1.81804e-13im -6.15023e-16-1.09078e-15im 7.91118e-16-1.04469e-15im
-1.37837e-16-3.0609e-9im 4.29751e-10+6.44627e-10im 4.29751e-10-6.44627e-10im -5.01384e-12-2.08911e-12im 5.01382e-12-2.08911e-12im -3.61552e-10+7.07385e-11im -3.61552e-10-7.07385e-11im 2.83376e-12-2.81014e-12im -2.83376e-12-2.81015e-12im -4.57901e-11+2.24071e-10im -4.57901e-11-2.24071e-10im -4.06243e-13-9.98407e-13im 4.06233e-13-9.98415e-13im 1.09292e-11+7.41901e-12im 1.09292e-11-7.41901e-12im -4.4566e-14-3.61759e-16im 4.45481e-14-3.97552e-16im 1.05374e-13-6.28694e-14im 1.01304e-13+7.23998e-14imWe can check its $L^2(\mathbb{S}^2)$ norm against an exact result:
norm(U⁰) ≈ sqrt(4π)trueSpin can be incremented by applying ð, either on the spin-$0$ coefficients:
U¹c = zero(U⁰)
for n in 1:N-1
U¹c[n, 1] = sqrt(n*(n+1))*U⁰[n+1, 1]
end
for m in 1:M÷2
for n in 0:N-1
U¹c[n+1, 2m] = -sqrt((n+m)*(n+m+1))*U⁰[n+1, 2m]
U¹c[n+1, 2m+1] = sqrt((n+m)*(n+m+1))*U⁰[n+1, 2m+1]
end
endor on the original function through analysis with spin-$1$ spherical harmonics:
F = [-(k[1]*(im*cospi(θ)*cospi(φ) + sinpi(φ)) + k[2]*(im*cospi(θ)*sinpi(φ)-cospi(φ)) - im*k[3]*sinpi(θ))*exp(im*(k⋅r(θ,φ))) for θ in θ, φ in φ]10×19 Matrix{ComplexF64}:
0.384531+0.240303im 0.405167+0.0811656im 0.376168-0.08059im 0.296802-0.228644im 0.17173-0.345547im 0.0121942-0.415058im -0.164084-0.425345im -0.334877-0.37218im -0.47719-0.260862im -0.571781-0.105926im -0.607006+0.0715576im -0.58067+0.248234im -0.499453+0.402825im -0.376425+0.519276im -0.227802+0.588046im -0.0700889+0.605739im 0.0817378+0.573795im 0.214909+0.497059im 0.318781+0.382786im
0.16375+0.41829im 0.162116+0.268361im 0.128477+0.110061im 0.056034-0.0418675im -0.0588234-0.167146im -0.211042-0.242529im -0.383203-0.247832im -0.546769-0.173942im -0.669506-0.0288607im -0.726395+0.162453im -0.708448+0.365492im -0.624684+0.547246im -0.496849+0.685091im -0.350631+0.769743im -0.208037+0.802572im -0.0835191+0.79034im 0.016198+0.740573im 0.0899356+0.659195im 0.13893+0.55045im
0.00394925+0.589002im -0.0517508+0.441892im -0.117739+0.283703im -0.202276+0.132736im -0.313608+0.0132761im -0.450683-0.0475101im -0.596088-0.0271995im -0.717144+0.0808626im -0.777618+0.259594im -0.754769+0.470906im -0.650847+0.67007im -0.491452+0.823236im -0.313141+0.917212im -0.149502+0.957306im -0.0230169+0.957535im 0.0563716+0.930505im 0.0891711+0.881918im 0.083394+0.810775im 0.0511118+0.713512im
-0.037706+0.764342im -0.167663+0.63523im -0.287398+0.487323im -0.400578+0.346904im -0.514827+0.244114im -0.62967+0.207393im -0.727882+0.254883im -0.777945+0.383583im -0.749353+0.563975im -0.631836+0.748617im -0.444017+0.892859im -0.224888+0.974245im -0.0160482+0.997806im 0.151723+0.986424im 0.261377+0.965157im 0.306018+0.948535im 0.286044+0.935214im 0.209798+0.910832im 0.0944472+0.857391im
0.0677134+0.90671im -0.132933+0.832114im -0.31272+0.722356im -0.463011+0.610267im -0.584223+0.531084im -0.67363+0.512373im -0.717075+0.564833im -0.692616+0.674959im -0.585463+0.805978im -0.402896+0.912201im -0.175697+0.960918im 0.0558086+0.947262im 0.258917+0.892857im 0.415705+0.832204im 0.518079+0.796763im 0.559103+0.802847im 0.530434+0.845158im 0.428907+0.898149im 0.265719+0.927327im
0.292353+0.944771im 0.0581069+0.961031im -0.162367+0.923101im -0.343726+0.861107im -0.474077+0.809852im -0.546787+0.794409im -0.553921+0.82068im -0.48849+0.87246im -0.353637+0.917742im -0.169005+0.924095im 0.0340849+0.875449im 0.226308+0.779766im 0.391603+0.66433im 0.527207+0.564551im 0.633493+0.513im 0.702424+0.529934im 0.714639+0.614543im 0.649544+0.740756im 0.502126+0.864782im
0.544556+0.82786im 0.335621+0.941682im 0.11692+0.992061im -0.074329+0.994789im -0.214433+0.975768im -0.291963+0.955971im -0.303796+0.942141im -0.254076+0.925583im -0.155653+0.88869im -0.0290943+0.815954im 0.10377+0.703491im 0.228799+0.562131im 0.344287+0.414327im 0.457269+0.288908im 0.572819+0.215991im 0.682616+0.220188im 0.761104+0.310327im 0.774782+0.470071im 0.701202+0.658896im
0.711777+0.57455im 0.575876+0.753741im 0.402359+0.877629im 0.227778+0.943189im 0.0807159+0.961489im -0.0227441+0.946886im -0.078234+0.908848im -0.0903981+0.849391im -0.0697694+0.765587im -0.0288206+0.65453im 0.0229208+0.517715im 0.0830404+0.363193im 0.157068+0.205921im 0.254332+0.067373im 0.379593-0.026087im 0.524118-0.0484074im 0.662039+0.0163403im 0.75688+0.163406im 0.777104+0.364037im
0.726602+0.264733im 0.673634+0.460754im 0.564658+0.62254im 0.42451+0.734172im 0.277783+0.792276im 0.142935+0.801596im 0.0303916+0.769731im -0.0562551+0.703542im -0.117442+0.608011im -0.154422+0.487008im -0.166521+0.34507im -0.149734+0.189474im -0.0974708+0.0321493im -0.00385489-0.109276im 0.131046-0.212499im 0.296212-0.254862im 0.468111-0.220723im 0.615102-0.109082im 0.707182+0.0630756im
0.597178-0.0185423im 0.601301+0.161421im 0.546217+0.32944im 0.442188+0.46657im 0.304261+0.559965im 0.149097+0.603283im -0.00741158+0.595734im -0.151367+0.540622im -0.271099+0.44408im -0.356905+0.314404im -0.400841+0.161998im -0.397025-0.000359286im -0.342634-0.157415im -0.239438-0.292068im -0.0953011-0.38711im 0.0751723-0.428079im 0.251638-0.406595im 0.410936-0.323077im 0.53139-0.187637imWe change plans with spin-$1$ now and reanalyze:
P = plan_spinsph2fourier(F, 1)
PA = plan_spinsph_analysis(F, 1)
U¹s = P\(PA*F)10×19 Matrix{ComplexF64}:
4.81048e-17+2.24152im 0.792499-0.528332im 0.792499+0.528332im -0.0752649+0.180636im 0.0752649+0.180636im -0.00429218-0.0219378im -0.00429218+0.0219378im 0.00124619+0.00125666im -0.00124619+0.00125666im -0.000105033-2.1464e-5im -0.000105033+2.1464e-5im 4.87035e-6-1.98165e-6im -4.87035e-6-1.98165e-6im -1.21684e-7+1.79257e-7im -1.21684e-7-1.79257e-7im 6.4556e-11-7.69602e-9im -6.42651e-11-7.6961e-9im 1.3734e-10+1.99137e-10im 1.25853e-10-2.02909e-10im
-0.725151+5.32221e-18im 0.361271+0.541907im -0.361271+0.541907im -0.0841092-0.0350455im -0.0841092+0.0350455im 0.00817504-0.00159947im -0.00817504-0.00159947im -0.000400574+0.000397236im -0.000400574-0.000397236im 6.06873e-6-2.9697e-5im -6.06873e-6-2.9697e-5im 5.08411e-7+1.24954e-6im 5.08411e-7-1.24954e-6im -4.23861e-8-2.87724e-8im 4.23861e-8-2.87724e-8im 1.69612e-9+1.41282e-11im 1.69628e-9-1.41119e-11im -4.07203e-11+2.96639e-11im 3.63017e-11+2.53159e-11im
-7.43762e-17-0.0844589im -0.145179+0.096786im -0.145179-0.096786im 0.00803495-0.0192839im -0.00803495-0.0192839im 0.000324556+0.00165884im 0.000324556-0.00165884im -7.29881e-5-7.36014e-5im 7.29881e-5-7.36014e-5im 5.02037e-6+1.02593e-6im 5.02037e-6-1.02593e-6im -1.9664e-7+8.00072e-8im 1.9664e-7+8.00072e-8im 4.25378e-9-6.26388e-9im 4.25378e-9+6.26388e-9im 6.53269e-12+2.1811e-10im -6.52146e-12+2.18144e-10im -1.63776e-11+3.366e-12im -1.53622e-11-3.76794e-12im
-0.000893686-1.84222e-18im -0.014106-0.0211589im 0.014106-0.0211589im 0.00272901+0.00113709im 0.00272901-0.00113709im -0.00022178+4.33918e-5im 0.00022178+4.33918e-5im 9.27716e-6-9.19981e-6im 9.27716e-6+9.19981e-6im -1.22275e-7+5.98394e-7im 1.22275e-7+5.98394e-7im -9.06826e-9-2.22464e-8im -9.06826e-9+2.22464e-8im 6.14926e-10+3.8178e-10im -6.14926e-10+3.81779e-10im -2.02946e-11+8.95683e-11im -2.03114e-11-8.95764e-11im -7.74583e-13-2.47296e-12im 9.78849e-13-2.19174e-12im
-2.77419e-17-0.00154725im 0.00179232-0.00119488im 0.00179232+0.00119488im -0.000108246+0.00025979im 0.000108246+0.00025979im -4.14499e-6-2.11852e-5im -4.14499e-6+2.11852e-5im 8.58374e-7+8.65614e-7im -8.58374e-7+8.65614e-7im -5.40698e-8-1.11423e-8im -5.40698e-8+1.11423e-8im 1.63049e-9-7.15813e-10im -1.63049e-9-7.15813e-10im 2.6e-10+2.34806e-10im 2.6e-10-2.34805e-10im -9.52905e-12-1.46897e-12im 9.53119e-12-1.47348e-12im 3.24098e-12-6.83905e-13im 3.02544e-12+8.12829e-13im
0.000232639-3.672e-17im 4.53837e-5+6.80755e-5im -4.53837e-5+6.80755e-5im -1.68233e-5-7.00961e-6im -1.68233e-5+7.00961e-6im 1.49733e-6-2.92965e-7im -1.49733e-6-2.92965e-7im -6.18845e-8+6.17602e-8im -6.18845e-8-6.17602e-8im 4.71847e-10-3.03855e-9im -4.71847e-10-3.03855e-9im 1.02475e-9-3.12162e-10im 1.02475e-9+3.12161e-10im -2.32851e-11+2.79721e-11im 2.3285e-11+2.79721e-11im 5.51359e-12-2.47804e-11im 5.51954e-12+2.47829e-11im 2.05758e-13+6.50666e-13im -2.63466e-13+5.82305e-13im
-2.539e-17+1.97929e-5im 3.54721e-6-2.36505e-6im 3.54721e-6+2.36505e-6im 2.74198e-7-6.5817e-7im -2.74198e-7-6.5817e-7im 1.49997e-8+7.82205e-8im 1.49997e-8-7.82205e-8im -2.72196e-9-2.73494e-9im 2.72196e-9-2.73494e-9im 4.22222e-10-1.58584e-9im 4.22222e-10+1.58584e-9im 1.4028e-11+4.2318e-11im -1.40281e-11+4.2318e-11im -7.00664e-11-6.32065e-11im -7.00664e-11+6.32064e-11im 1.8132e-12+2.78513e-13im -1.81361e-12+2.79436e-13im -8.89729e-13+1.90111e-13im -8.28862e-13-2.3028e-13im
-1.0618e-6-1.60227e-16im 5.06265e-7+7.59331e-7im -5.06265e-7+7.59331e-7im 2.28542e-9+1.27481e-9im 2.28542e-9-1.27481e-9im -1.53825e-9+3.35011e-10im 1.53825e-9+3.35011e-10im -2.24932e-9-2.42853e-9im -2.24932e-9+2.42853e-9im 1.43487e-10+3.57671e-11im -1.43487e-10+3.57671e-11im -4.84249e-10+1.47683e-10im -4.84249e-10-1.47683e-10im 9.17835e-12-1.10319e-11im -9.17827e-12-1.10319e-11im -2.61497e-12+1.18851e-11im -2.61825e-12-1.18864e-11im -8.21156e-14-2.58948e-13im 1.05894e-13-2.32289e-13im
1.80271e-18-2.92736e-8im -6.36569e-8+4.17996e-8im -6.36569e-8-4.17996e-8im 6.18282e-10-1.39021e-9im -6.18282e-10-1.39021e-9im -4.16445e-10+7.77363e-11im -4.16445e-10-7.77363e-11im 8.6631e-12-8.2083e-12im -8.66306e-12-8.20835e-12im -6.71953e-11+2.52391e-10im -6.71953e-11-2.52391e-10im -1.00987e-12-3.04635e-12im 1.00985e-12-3.0463e-12im 1.11584e-11+1.00647e-11im 1.11584e-11-1.00647e-11im -1.30587e-13-2.0048e-14im 1.30633e-13-2.01267e-14im 1.41977e-13-3.03707e-14im 1.32247e-13+3.68603e-14im
-1.3507e-9+1.40212e-16im -3.62349e-10-4.74615e-10im 3.62349e-10-4.74615e-10im 1.35757e-9-2.98609e-9im 1.35757e-9+2.98609e-9im -2.68037e-12-1.17672e-11im 2.68035e-12-1.17672e-11im 3.06502e-9+3.30916e-9im 3.06502e-9-3.30916e-9im -2.9668e-11-7.39519e-12im 2.9668e-11-7.39519e-12im 6.60325e-10-2.01412e-10im 6.60325e-10+2.01411e-10im -1.89842e-12+2.28193e-12im 1.89842e-12+2.28191e-12im 3.56751e-12-1.6238e-11im 3.57202e-12+1.62398e-11im 1.69962e-14+5.36059e-14im -2.19325e-14+4.80988e-14imFinally, we check $L^2(\mathbb{S}^2)$ norms against another exact result:
norm(U¹c) ≈ norm(U¹s) ≈ sqrt(8π/3*(k⋅k))trueThis page was generated using Literate.jl.