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Helmholtz decompositionHeat equationLaplace's equationLaplace operatorHecke algebraGraph fourier transformEigenvalue perturbationEuler's identityHarmonic functionDifferential equationLiouville's equationInverse scattering transformSommerfeld radiation conditionEuler's theoremErnst equationIntegro-differential equationSpectrum of a matrixLaplace's methodPoisson's equationD'alembert's equationEigenfunctionPartial differential equationEuler's formulaComplementary functionHessian matrixElliptic operatorBiharmonic equationFredholm integral equationBernoulli differential equationLaguerre polynomialKoenigs functionIntegral equationEuler classFredholm theoryAlgebraic equationHankel functionMethod of characteristicsModeMathieu functionLaguerre's equationSpectrumTwo-sided laplace transformDiscrete poisson equationGauss-seidel methodHill differential equationMellin transformCritical functionAbel's theoremParabolic cylinder functionHellinger integralHomotopy analysis methodAssociated legendre polynomialsBeltrami identityLaplace distributionDifferential operatorLaplace transformJensen's inequalityHamiltonian vector fieldLaplace-beltrami operatorHermite normal formSchinzel's hypothesis hHamiltonianComplex analysisPentagonal number theoremGeometric multiplicitySpectral radiusLagrangian functionJacobian matrix and determinantEigenmatrixDiophantine geometryOperator theoryEigengapDirichlet energyLemniscate of bernoulliBounded inverse theoremInverse iterationMatrix exponentialClosed-subgroup theoremFredholm operatorFrobenius inner productJacobian conjectureBeta functionConvex conjugatePoisson summation formulaFenchel's duality theoremEuler methodHodge decompositionHilbert transformEigendecomposition of a matrixHilbert manifoldEuler's criterionMilstein methodRiccati equationImplicit surfaceLagrange inversion theoremDiscrete laplace operatorLaplace-de rham operatorLoewner orderEuler summationHausdorff paradox