Point sources and complex Gaussian kernels

Reconstructed by GPT-6 Astra (OpenAI), Ultra reasoning effort, 4 October 2026. Public domain (CC0).

A fundamental solution is normalized by its point source. For the Laplacian, a small enclosing flux determines that normalization. For complex diffusion, it is determined by a Gaussian mass limit and a continuously chosen matrix phase. The determinant's numerical value alone does not determine that phase.

The exact preceding proofs used here are Boundary flux and weak identities, Theorem 2.1; Homogeneous extensions and angular moments, Theorem 6.1 and its angular foundations, A4–A5; the Schwartz Fourier foundations, F1–F5; Cauchy kernels and boundary limits, Corollary 2.2; Holomorphic boundaries in convex cones, the polydisk proof and Lemma 3.1; and the compact approximation in Order, positivity and distributional limits. The finite algebra foundation, Sections 10.1–10.6, proves the determinant, inverse and orthogonal-basis operations. We prove the additional diagonalization, determinant differentiation and square-integral arguments below.

Fix the sign of a point source

For a constant-coefficient operator , a fundamental solution is a distribution with . Adding a distribution with gives another one; the definition alone specifies neither uniqueness nor support.

Write , and let be the Euclidean area of from U018.

Theorem 1.1 (the positive Laplacian point source). The locally integrable functions define distributions satisfying . For , their distributional first derivatives are the locally integrable functions

Proof. For , polar integration bounds the integral near zero of by a constant times . In dimension two the relevant integral is , evaluated by integration by parts or the logarithmic integral in the gamma companion. Ordinary differentiation off zero gives (1.2), whose absolute radial integral near zero is bounded by a constant times .

It remains to verify that differentiation has added no point term. Given a compact smooth , integrate the smooth product on the annulus , with outside its support, using U011's flux theorem in the -th coordinate. The outer boundary contribution is zero. The inner contribution has modulus at most . For this is ; for it is . Both tend to zero. The volume integrals converge by the local integrability just proved. Thus integration by parts gives exactly (1.2) on every test.

U018, (6.5), proves by its complete radial test and flux calculation. Taking the divergence of (1.2) therefore gives . For , split into the two half-lines. Integration by parts on each gives ; the boundary terms at infinity vanish by compact support. Their sum is . This covers every dimension.

For , with and , (1.2) gives as distributions, not only off zero. The identity consequently agrees with the Cauchy point-source normalization proved in U013.

Corollary 1.2 (zeros and poles as signed sources). If is a meromorphic function, not identically zero, on a connected planar open set , then is locally integrable and Meromorphic means locally a quotient of holomorphic functions with denominator not identically zero. The order is positive for a zero, negative for a pole, and zero elsewhere. The nonzero terms in this sum are locally finite.

Proof. The power-series and identity principles are supplied by U013 and U015. They imply that a holomorphic function which is not identically zero on a connected disk has isolated zeros: at a zero, take its first nonzero Taylor coefficient and factor out the corresponding power. If all coefficients vanished, the function would vanish on a disk and hence on the connected domain.

No local numerator for can be identically zero. To see this, let be the set where vanishes identically in a neighborhood. This set is open. At a limit point of , choose a quotient on a connected disk, with not identically zero. The nonempty open part of in that disk meets the set , since cannot vanish on an open subset. Thus vanishes on an open set, and the identity principle gives on the disk. This shows that is also closed. Connectedness and the hypothesis on force to be empty.

Factoring the first nonzero Taylor powers of the local numerator and denominator now gives , where and is holomorphic and nonzero on a small disk. Each such disk has at most its center as a zero or pole. A finite subcover of a compact subset of proves local finiteness.

For completeness, is harmonic without assuming a logarithm theorem. Shrink the disk until has modulus less than . The series and all its derivatives converge uniformly on smaller disks; the scalar series theorem proves . Differentiating shows it is one, by its value at zero and the segment fundamental theorem. Therefore , using from the exponential laws. Choose a real argument with , using the unit-circle parametrization proved in the scalar foundation. Then is a holomorphic logarithm of : the preceding exponential identity gives . Its real part is . The real part of a smooth holomorphic function is harmonic by the Cauchy–Riemann equations and commutation of mixed derivatives.

Thus is locally integrable. Theorem 1.1 gives its local Laplacian . At a nonzero ordinary point it is harmonic. These identities agree on overlaps. A finite smooth partition on the support of a test sums them, proving (1.5) globally. Such compact partitions are supplied in the scalar/test foundations included with U011.

U018 A5 and its polar formula give the constants The first identity uses the fully proved Gaussian mass and the Euler gamma integral; the second is . The gamma companion proves , , and . Finite iteration therefore yields, for , The odd-dimensional expressions also hold at , with : the sphere has two points and the unit interval has length two.

Choose the Gaussian phase from its matrix

We first supply the two finite matrix facts used in the construction.

Real symmetric diagonalization. A real symmetric matrix has a real orthonormal eigenbasis. Indeed, the continuous quadratic form has a maximum at some unit vector , by compactness of the real unit sphere. For , differentiate at the value of this form at . Its derivative is , which must vanish at a maximum. Thus is orthogonal to every , hence . Symmetry makes invariant: . Choose an orthonormal basis there by the finite Gram construction and repeat on this smaller symmetric matrix. Induction, starting at dimension one, proves the assertion. Positivity of is equivalent to positivity of all its eigenvalues by the resulting diagonal quadratic form. In particular a positive definite has a positive lower bound .

Determinant differentiation. If is differentiable and invertible, the finite multilinear determinant expansion gives To check the coefficient, write . Multilinearity gives : the identity permutation contributes the stated linear terms, while any nonidentity permutation uses at least two off-diagonal entries. Multiplicativity of the determinant then proves (M1). Determinants are finite polynomials and inverses are the adjugate divided by the determinant, as proved in the algebra foundation; in particular inversion is continuous and holomorphic wherever the determinant is nonzero.

Let be complex symmetric. Its entrywise real part equals its Hermitian part . For a complex vector , the real symmetric form satisfies . Hence positivity on real vectors is equivalent to Hermitian positivity on complex vectors. Set The cone is convex and open in the independent complex symmetric entries. Openness follows from the positive minimum of on the unit sphere and the elementary finite-coordinate bound for a small matrix perturbation. Every is invertible: would give , impossible for .

Lemma 2.1 (the determinant branch). On there is a unique holomorphic with , positive on real positive definite matrices. It extends continuously, and by a holomorphic function in a neighborhood of each point, to . On this extended set, For real symmetric invertible , where is the number of positive eigenvalues minus the number of negative ones.

Proof. Define and For , has positive definite real part for every , so it is invertible. On a compact subset of , its determinant is bounded away from zero uniformly in ; the rational inverse formula then bounds every matrix-entry derivative uniformly. Difference quotients under this finite integral prove that is jointly continuous and coordinate holomorphic. U015's polydisk proof supplies its full holomorphy and smoothness.

For fixed , (M1) shows that has derivative zero and value one at zero. This direct integrating-factor calculation gives . Set . For real positive definite , the proved orthogonal diagonalization turns into : integrate for each positive eigenvalue. Therefore is positive. Any other branch has a continuous ratio to with square one. Its ratio is constant on the convex connected cone, and is one at . This proves uniqueness.

If and is invertible, has strictly positive real part for and an invertible endpoint at . The continuous nonzero determinant on this compact segment has positive minimum modulus. Uniform continuity gives a neighborhood of on which every stays invertible. Formula (2.4) therefore defines a holomorphic extension there by the same rational bounds. The formulas coincide wherever both are defined, and their values on the semidefinite set are their limits along . This proves the stated continuous and local holomorphic extension.

Transposing both inverse identities gives , so the inverse is symmetric. Matrix multiplication gives Indeed the right side is half the sum . Thus inversion preserves the two domains. On , is continuous with square one and equals one at , proving the first identity of (2.2). For fixed , the same reasoning applies to . Approximation by , continuity of inversion and the local extension prove both identities on the boundary as well.

Finally diagonalize with real orthogonal . Uniqueness gives on , and continuity gives it on the boundary. On diagonal matrices in , the branch is the product of scalar square roots on the right half-plane: the product is holomorphic, squares to the determinant, and is positive on positive real diagonals. The scalar right-half-plane logarithm, proved in U017, has argument tending to or at a nonzero positive or negative imaginary number. Consequently the -th factor at tends to . Their product gives (2.3).

For example , but the branch prescribed by its matrix path is .

A bounded Fourier multiplier supplies the mass limit

Use the conventions proved in the Schwartz Fourier companion: The transform is a continuous automorphism of the Schwartz space, and its complex-linear transpose defines the transform on tempered distributions. In particular, bounded functions define tempered distributions, since Schwartz tests are integrable.

Theorem 3.1 (complex Gaussian transform and concentration). For and , the bounded smooth function has the tempered Fourier transform Its normalized pairing converges to as . This holds for Schwartz tests and for every compact test with integer . Uniformly in and , In particular, every even integer is allowed.

Proof of the transform. Start with and real positive definite . The orthogonal diagonalization proved above reduces the Fourier integral to the product of scalar Gaussian integrals. The real linear substitution has absolute determinant one, by the affine integration theorem included with U011. The scalar formula is proved with its exact mass in the Fourier companion, F3: The product yields (3.2) with .

For fixed real , both sides are holomorphic in the independent symmetric entries on . On a compact subset of this cone, compactness of its product with the real unit sphere gives one with . Each matrix-entry derivative of the integrand is a polynomial in times that integrand. A polynomial times is integrable, by the scalar Gaussian estimates in F3 and Fubini. Dominated difference quotients therefore give joint continuity and coordinate holomorphy, hence full holomorphy by U015's polydisk proof. The branch and rational inverse give it for the right side too.

Equality on the real positive definite matrices suffices here. Around a fixed such matrix, choose a small complex polydisk inside whose real coordinate cube consists of positive definite matrices. Hold all but the first entry real, and use the one-variable identity principle on the real interval of that first entry. Repeat coordinate by coordinate, now allowing the previously treated entries to be complex. Equality follows throughout the polydisk. U015, Lemma 3.1, then extends it across the connected cone. The substitution , with Jacobian , proves the formula for every .

For a boundary matrix take . Since , the original exponential has modulus at most one; dominated convergence against each Schwartz test gives its tempered limit. Formula (2.5) gives the same bound for the transformed exponential. The constants and the inverse matrices converge, so the transformed expressions also converge against each Schwartz test. The Fourier transpose is continuous for this convergence, by F5. Passing to the limit proves (3.2) for every .

Proof of concentration and the uniform bound. The inverse transpose formula for the even multiplier in (3.2) gives Precisely, if , then is the displayed exponential. The inverse test transform is . Changing to and using evenness gives (3.5), including its sign and factor. Its multiplier has modulus at most one by (2.5), and tends pointwise to one as . Since , dominated convergence and inversion at zero prove the concentration for Schwartz tests, including compact smooth ones.

We supply the square-integral step needed to reduce the regularity requirement. For , insert (3.1) for in . Absolute Fubini is valid because . The inner integral is , and hence . This is only a Schwartz identity and uses no completed Fourier theory.

The weight is integrable when . Indeed U018's polar formula reduces the tail to a constant times ; on the unit ball the weight is bounded. Cauchy–Schwarz gives For the second inequality expand The Fourier derivative identity and the preceding square-integral formula identify each weighted square integral with . Taking a square root and bounding the finite Euclidean norm by the sum proves (3.6). Finally . Together with (3.5) this proves (3.4), with a constant independent of .

For compact tests, the U008 approximation has a common compact support and converges in every derivative through order . On that support it also converges in the norms in (3.4). The estimate makes the normalized pairings converge uniformly over and as the smooth approximations converge. At each fixed , their ordinary integrals in (3.3) converge to the displayed integral by uniform convergence on that compact support. Passing the uniform estimate, then using the smooth concentration limit and a three-term difference estimate, proves both (3.4) and the mass limit for the test.

There is no need for the real and imaginary parts to commute. For example Its determinant is , and the displayed real parts are semidefinite. Formula (3.5) still has a multiplier of modulus at most one, even though the original Gaussian lacks decay in one real direction. Purely imaginary invertible matrices are included for the same reason.

Integrate the kernels in time

Let , , and . For define It is zero for negative time. The time-zero distribution is defined by the iterated pairing in the next theorem.

Theorem 4.1 (heat and complex diffusion point sources). For every compact smooth space-time test, converges and defines a distribution supported in , of order at most every integer . It satisfies For real positive definite , it is locally integrable and smooth off the space-time origin. In particular is a fundamental solution of .

Proof. Put . Formula (2.5) shows that , and (2.2) gives . Thus . For tests on one compact space-time support, (3.4) bounds this uniformly by a fixed norm times a constant depending on that support; all spatial norms are bounded by its finite enclosing box volume times sup norms. The function is continuous for , by differentiation under the compact spatial integral on intervals bounded away from zero. Compact time support and the uniform bound make (4.2) absolutely integrable as a scalar function of time. They prove the finite-order distribution estimate. Tests supported in negative time pair to zero.

Formula (3.2) gives For the classical differential identity when , put . Then and Contracting with uses symmetry, , and . It gives . This verifies the equation directly with every coefficient.

For , spatial integration by parts is valid because the test is compact and the kernel is smooth. It combines with the last identity and time integration to give Here each denotes spatial pairing. Equivalently the integrand on the left is minus the derivative of , and the upper time endpoint vanishes by the test support. The left side converges to , by the established existence for differentiated tests. The difference between the right side tested at and at tends to zero by (3.4) and convergence in the fixed compact norm. The concentration theorem gives the remaining limit , proving (4.3).

When is real positive definite, diagonalization and the scalar Gaussian integrals show and . Tonelli bounds its absolute integral on a compact spatial set and by , proving local integrability through time zero.

On a compact spatial set away from zero, positivity of gives . Inductively every space-time derivative of (4.1) is a finite polynomial in times the same kernel, and is bounded there by . This tends to zero faster than any power of , since the exponential series bounds below by every chosen power of . Consequently each derivative tends uniformly to zero as . Extending by zero for is smooth: induct on derivative order, using the fundamental theorem in a coordinate segment to identify each derivative across ; for the time derivative the difference quotient is bounded by the next positive-time derivative, which tends to zero. Positive-time smoothness was already explicit. Thus the extension is smooth at every point except possibly .

For general complex matrices the theorem asserts the distribution and its source; smooth continuation across time zero away from the origin was proved only for real positive definite matrices.

With the Schrödinger convention , the supported fundamental solution is Multiplying the unit-source identity for by proves the normalization. The matrix branch supplies the separate phase .