Generalized Gaussian Integrals

• Generalized Gaussian integrals are extensions of classical Gaussian formulas featuring modified exponents, measures, and integration domains.
• They are evaluated using techniques like completion of the square, analytic continuation, and subtracted singular term regularization.
• These integrals have wide applications in quantum mechanics, random matrix theory, and computational methods across physics and mathematics.

Searching arXiv for recent and foundational papers on generalized Gaussian integrals and closely related formulations. arXiv search query: generalized Gaussian integrals quantum parameter finite dimension Gaussian integral generalized gamma function Gaussian-like integrals Generalized Gaussian integrals are extensions of the classical Gaussian integral in which one or more of the defining features of the standard model are modified: the exponent may remain quadratic but become matrix-valued, parameter-dependent, oscillatory, or geodesic; the measure may be deformed by symmetry or regularization; the domain may be finite, manifold-valued, or singular at an endpoint; and the integrand may be augmented by polynomial, special-function, or source terms. In the recent literature, the phrase covers several distinct but overlapping constructions, including finite-dimensional $\hbar$-dependent quadratic integrals, moment hierarchies generated by derivatives, generalized integrals defined by subtraction of singular homogeneous terms, and non-Euclidean Gaussian partition functions on symmetric spaces (Camosso, 2021, Dereziński et al., 2023, Heuveline et al., 2021, Pant et al., 11 Aug 2025).

1. Classical kernel and the baseline formulas

The point of departure is the one-dimensional Gaussian identity

$\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi},$

together with its standard derivation by squaring the integral and converting the resulting integral over $\mathbb R^2$ to polar coordinates. This immediately yields the rescaled formula

$$\int_{-\infty}^{+\infty} e^{-a x^2}\,dx=\sqrt{\frac{\pi}{a}},\qquad a>0,$$

and, after completion of the square,

$$\int_{-\infty}^{+\infty} e^{-a x^2-bx}\,dx =\sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}},$$

which is the elementary prototype for essentially all higher-dimensional and parameter-dependent Gaussian evaluations (Camosso, 2021).

Classical moment formulas are equally foundational. The even moments satisfy

$$\int_{-\infty}^{+\infty} x^{2n}e^{-x^2}\,dx =\frac{(2n)!}{2^{2n}n!}\sqrt{\pi},$$

while odd moments vanish by symmetry. This parity structure persists under rescaling and under matrix generalization, and later reappears as Wick pairing in multivariate settings (Camosso, 2021).

A separate classical issue is the finite-boundary Gaussian integral

$$P(t)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}e^{-x^2/2}\,dx,$$

equivalently the symmetric normal probability $P(t)=2\Phi(t)-1$ or an error-function evaluation. The finite-boundary problem does not admit an elementary antiderivative, and one line of work replaces it by systematically refinable approximations of the form

$$P(t)\approx \sqrt{1-\frac1N\sum_{n=1}^N e^{-k_n^2 t^2/2}},$$

with $k_n\in[1,\sqrt2]$ obtained from binary or ternary geometric partitioning (Martila et al., 2022). This finite-boundary setting should be distinguished from the exact whole-line Gaussian formulas.

2. Quadratic forms, matrices, and generating-function structure

In finite dimension, the canonical multivariate generalization replaces $\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi},$0 by a positive definite quadratic form. For a real symmetric positive definite matrix $\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi},$1,

$\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi},$2

obtained by diagonalizing $\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi},$3 and reducing the integral to a product of one-dimensional Gaussians (Camosso, 2021). The same framework yields the Gaussian Fourier-transform identity: if $\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi},$4 with $\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi},$5, then $\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi},$6 is again Gaussian with inverse matrix $\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi},$7.

Moments of multivariate Gaussians are generated by the source-dependent partition function

$\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi},$8

Differentiation in $\int_{-\infty}^{+\infty} e^{-x^2}\,dx=\sqrt{\pi},$9 recovers correlation tensors. In particular,

$\mathbb R^2$0

and the higher-order derivatives satisfy the pairing formula identified with Wick’s theorem, where the $\mathbb R^2$1-point moment is expressed as a sum over pairings of entries of $\mathbb R^2$2 (Camosso, 2021).

The same finite-dimensional logic extends to matrix-valued variables. For Hermitian matrices $\mathbb R^2$3, the quadratic form $\mathbb R^2$4 turns $\mathbb R^2$5 into a Gaussian integration space, and the normalized Gaussian measure

$\mathbb R^2$6

leads to

$\mathbb R^2$7

This is still a Gaussian integral in the strict quadratic-form sense; only the ambient vector space has changed (Camosso, 2021).

3. Quantum-scale dependence and semiclassical finite-dimensional models

A prominent recent generalization introduces a formal quantum parameter $\mathbb R^2$8, or equivalently $\mathbb R^2$9, in the exponent. The basic model is

$$\int_{-\infty}^{+\infty} e^{-a x^2}\,dx=\sqrt{\frac{\pi}{a}},\qquad a>0,$$0

with $$\int_{-\infty}^{+\infty} e^{-a x^2}\,dx=\sqrt{\frac{\pi}{a}},\qquad a>0,$$1 in the introductory discussion and with $$\int_{-\infty}^{+\infty} e^{-a x^2}\,dx=\sqrt{\frac{\pi}{a}},\qquad a>0,$$2 typically quadratic. In explicit finite-dimensional examples the parameter is written directly as $$\int_{-\infty}^{+\infty} e^{-a x^2}\,dx=\sqrt{\frac{\pi}{a}},\qquad a>0,$$3, giving exact formulas such as

$$\int_{-\infty}^{+\infty} e^{-a x^2}\,dx=\sqrt{\frac{\pi}{a}},\qquad a>0,$$4

and

$$\int_{-\infty}^{+\infty} e^{-a x^2}\,dx=\sqrt{\frac{\pi}{a}},\qquad a>0,$$5

obtained by translations and completion of the square (Camosso, 2021). These formulas are exact finite-dimensional analogues of heat-kernel composition and Gaussian propagation.

The same paper treats a more elaborate oscillatory kernel from geometric quantization,

$$\int_{-\infty}^{+\infty} e^{-a x^2}\,dx=\sqrt{\frac{\pi}{a}},\qquad a>0,$$6

for which the corresponding integral over $$\int_{-\infty}^{+\infty} e^{-a x^2}\,dx=\sqrt{\frac{\pi}{a}},\qquad a>0,$$7 reduces to a determinant of a complex quadratic form and produces a prefactor $$\int_{-\infty}^{+\infty} e^{-a x^2}\,dx=\sqrt{\frac{\pi}{a}},\qquad a>0,$$8 (Camosso, 2021). Here the Gaussian is no longer purely decaying; oscillation is encoded by the complex parameter $$\int_{-\infty}^{+\infty} e^{-a x^2}\,dx=\sqrt{\frac{\pi}{a}},\qquad a>0,$$9.

A standard misconception is that these formulas amount to a rigorous path-integral theory. The finite-dimensional treatment explicitly restricts itself to exact Gaussian integrals and formal perturbative expansions. The perturbative model

$$\int_{-\infty}^{+\infty} e^{-a x^2-bx}\,dx =\sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}},$$0

is handled by expanding $$\int_{-\infty}^{+\infty} e^{-a x^2-bx}\,dx =\sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}},$$1 into a series and reducing the coefficients to derivatives of the Gaussian generating function in the source $$\int_{-\infty}^{+\infty} e^{-a x^2-bx}\,dx =\sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}},$$2 (Camosso, 2021). The construction is therefore semiclassical and finite-dimensional by design, not a fully rigorous infinite-dimensional functional integral.

4. Broader meanings of “generalized Gaussian”

Not every generalized Gaussian integral remains quadratic. One frequently studied family is

$$\int_{-\infty}^{+\infty} e^{-a x^2-bx}\,dx =\sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}},$$3

which reduces to $$\int_{-\infty}^{+\infty} e^{-a x^2-bx}\,dx =\sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}},$$4 at $$\int_{-\infty}^{+\infty} e^{-a x^2-bx}\,dx =\sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}},$$5 and behaves asymptotically like $$\int_{-\infty}^{+\infty} e^{-a x^2-bx}\,dx =\sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}},$$6 after inserting the Laurent expansion of $$\int_{-\infty}^{+\infty} e^{-a x^2-bx}\,dx =\sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}},$$7 at $$\int_{-\infty}^{+\infty} e^{-a x^2-bx}\,dx =\sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}},$$8 (Pant et al., 11 Aug 2025). In this usage, “generalized Gaussian” means replacement of the quadratic exponent by a higher power.

A closely related moment-based generalization is the even-moment family

$$\int_{-\infty}^{+\infty} e^{-a x^2-bx}\,dx =\sqrt{\frac{\pi}{a}}\,e^{\frac{b^2}{4a}},$$9

for which the coefficient pattern is encoded by the derivatives of

$$\int_{-\infty}^{+\infty} x^{2n}e^{-x^2}\,dx =\frac{(2n)!}{2^{2n}n!}\sqrt{\pi},$$0

The parameter

$$\int_{-\infty}^{+\infty} x^{2n}e^{-x^2}\,dx =\frac{(2n)!}{2^{2n}n!}\sqrt{\pi},$$1

generates the sequence $$\int_{-\infty}^{+\infty} x^{2n}e^{-x^2}\,dx =\frac{(2n)!}{2^{2n}n!}\sqrt{\pi},$$2, yielding the classical closed form

$$\int_{-\infty}^{+\infty} x^{2n}e^{-x^2}\,dx =\frac{(2n)!}{2^{2n}n!}\sqrt{\pi},$$3

In this framework the generalized content lies not in the exponent but in the systematic extraction of Gaussian moments from a single Taylor series (Sales et al., 2018).

Another extension leaves the Gaussian kernel exponential but replaces the quadratic form by a positive homogeneous function. If $$\int_{-\infty}^{+\infty} x^{2n}e^{-x^2}\,dx =\frac{(2n)!}{2^{2n}n!}\sqrt{\pi},$$4 is $$\int_{-\infty}^{+\infty} x^{2n}e^{-x^2}\,dx =\frac{(2n)!}{2^{2n}n!}\sqrt{\pi},$$5-homogeneous with weights $$\int_{-\infty}^{+\infty} x^{2n}e^{-x^2}\,dx =\frac{(2n)!}{2^{2n}n!}\sqrt{\pi},$$6, then

$$\int_{-\infty}^{+\infty} x^{2n}e^{-x^2}\,dx =\frac{(2n)!}{2^{2n}n!}\sqrt{\pi},$$7

A concrete consequence is

$$\int_{-\infty}^{+\infty} x^{2n}e^{-x^2}\,dx =\frac{(2n)!}{2^{2n}n!}\sqrt{\pi},$$8

which returns the ordinary Gaussian at $$\int_{-\infty}^{+\infty} x^{2n}e^{-x^2}\,dx =\frac{(2n)!}{2^{2n}n!}\sqrt{\pi},$$9 (Camosso, 2021).

The same broadening occurs in weighted gamma-beta constructions. The two-dimensional generalized gamma function

$$P(t)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}e^{-x^2/2}\,dx,$$0

factorizes as a generalized beta factor times a generalized gamma factor,

$$P(t)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}e^{-x^2/2}\,dx,$$1

so the ratio variable $$P(t)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}e^{-x^2/2}\,dx,$$2 carries the beta structure and $$P(t)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}e^{-x^2/2}\,dx,$$3 carries the gamma structure (Ponomarenko, 2024). This is not Gaussian in the strict quadratic sense, but it belongs to the same exponential-integral family and is used to represent oscillatory double integrals with kernels $$P(t)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}e^{-x^2/2}\,dx,$$4 and $$P(t)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}e^{-x^2/2}\,dx,$$5 (Ponomarenko, 2024).

5. Generalized integrals as regularization and anomaly

In another established usage, a generalized Gaussian integral is not a new exponent at all, but a regularized linear functional extending ordinary integration to functions with finitely many non-integrable homogeneous singular terms at an endpoint. If near $$P(t)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}e^{-x^2/2}\,dx,$$6,

$$P(t)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}e^{-x^2/2}\,dx,$$7

the generalized integral is defined by subtracting the singular terms, integrating the remainder, and restoring the regularized monomial contributions (Dereziński et al., 2023). For a monomial on $$P(t)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}e^{-x^2/2}\,dx,$$8,

$$P(t)=\frac{1}{\sqrt{2\pi}}\int_{-t}^{t}e^{-x^2/2}\,dx,$$9

This regularization is linear and agrees with the ordinary integral whenever the latter converges, but it is not fully scale invariant. Under $P(t)=2\Phi(t)-1$0,

$P(t)=2\Phi(t)-1$1

The logarithmic correction is the anomaly; it vanishes exactly when the coefficient of $P(t)=2\Phi(t)-1$2 vanishes (Dereziński et al., 2023). This failure of naive scaling invariance is essential rather than incidental, especially at integer parameter values.

The main special-function applications concern bilinear integrals of Macdonald and Gegenbauer functions. Ordinary integrals such as

$P(t)=2\Phi(t)-1$3

or their Gegenbauer analogues converge only in restricted parameter ranges, typically $P(t)=2\Phi(t)-1$4. Generalized integration extends these formulas by analytic continuation, and at integer $P(t)=2\Phi(t)-1$5 produces explicit logarithmic, digamma, and finite-sum correction terms (Dereziński et al., 2023, Dereziński et al., 2023).

This regularized framework has direct operator-theoretic content. Macdonald functions encode Euclidean Green functions, while Gegenbauer functions govern Green functions on the sphere and hyperbolic space. In dimensions $P(t)=2\Phi(t)-1$6, point-potential perturbations of the Laplacian produce genuine self-adjoint operators; in higher dimensions they do not. Generalized integrals nevertheless define a generalized $P(t)=2\Phi(t)-1$7 and hence a generalized Green function in arbitrary dimension (Dereziński et al., 2023). A plausible implication is that generalized Gaussian integration here plays a renormalization-like role: divergent scalar products are replaced by finite, scheme-sensitive but explicit quantities.

6. Geometric and computational realizations

On Riemannian manifolds, Gaussian distributions are obtained by replacing the Euclidean norm by geodesic distance: $P(t)=2\Phi(t)-1$8 For general manifolds the partition function $P(t)=2\Phi(t)-1$9 depends nonlinearly on the center $$P(t)\approx \sqrt{1-\frac1N\sum_{n=1}^N e^{-k_n^2 t^2/2}},$$0, but on a Riemannian symmetric space $$P(t)\approx \sqrt{1-\frac1N\sum_{n=1}^N e^{-k_n^2 t^2/2}},$$1 it becomes independent of $$P(t)\approx \sqrt{1-\frac1N\sum_{n=1}^N e^{-k_n^2 t^2/2}},$$2 and reduces to a finite-dimensional Cartan integral

$$P(t)\approx \sqrt{1-\frac1N\sum_{n=1}^N e^{-k_n^2 t^2/2}},$$3

This is a genuinely non-Euclidean generalized Gaussian integral (Heuveline et al., 2021).

For the cone of positive definite Hermitian matrices, the symmetric-space partition function becomes a $$P(t)\approx \sqrt{1-\frac1N\sum_{n=1}^N e^{-k_n^2 t^2/2}},$$4 log-normal random-matrix integral and is exactly computable through the Stieltjes–Wigert polynomials. The exact formula

$$P(t)\approx \sqrt{1-\frac1N\sum_{n=1}^N e^{-k_n^2 t^2/2}},$$5

connects generalized Gaussian normalization to random matrices and to $$P(t)\approx \sqrt{1-\frac1N\sum_{n=1}^N e^{-k_n^2 t^2/2}},$$6 Chern–Simons theory on $$P(t)\approx \sqrt{1-\frac1N\sum_{n=1}^N e^{-k_n^2 t^2/2}},$$7 (Heuveline et al., 2021). In the real SPD and Siegel cases exact finite-$$P(t)\approx \sqrt{1-\frac1N\sum_{n=1}^N e^{-k_n^2 t^2/2}},$$8 methods are less tractable, and the analysis moves to large-$$P(t)\approx \sqrt{1-\frac1N\sum_{n=1}^N e^{-k_n^2 t^2/2}},$$9 saddle-point theory.

Computational quantum chemistry and mathematical physics provide another major application domain. In two dimensions, primitive Cartesian Gaussian type orbitals reduce many overlap and operator integrals to one-dimensional recursions, while Coulomb attraction and interaction integrals are expressed through exponentially scaled modified Bessel functions $k_n\in[1,\sqrt2]$0 and McMurchie–Davidson-like recursions (Schøyen et al., 2021). For Gaussian-damped triple spherical Bessel integrals,

$k_n\in[1,\sqrt2]$1

closed forms are obtained in terms of incomplete Gamma functions and regularized hypergeometric functions, either recursively or by a non-recursive Hankel–Bowman method (Chellino et al., 2023).

A related transform theory for quantum amplitudes replaces the usual Gaussian auxiliary exponential by a Macdonald-function kernel. For products of Slater orbitals, this alternative transform uses one fewer auxiliary integration than the Gaussian transform, though at the cost of more complicated $k_n\in[1,\sqrt2]$2-kernels with square-root arguments (Straton, 2022). This suggests that, in applied settings, generalized Gaussian integration is often less a single formula than a strategy: complete the square when possible, otherwise convert the quadratic structure into a special-function kernel that preserves analytic control.

The modern literature therefore supports no single canonical definition of generalized Gaussian integrals. The common core is structural rather than terminological: exact evaluation or controlled extension of Gaussian-type expressions through translation, diagonalization, source differentiation, analytic continuation, regularization of endpoint singularities, or symmetry reduction. Under that broad heading, the subject now spans finite-dimensional semiclassical analysis, generalized moment theory, endpoint-regularized special-function bilinear forms, non-Euclidean probability, random matrices, and computational integral transforms (Camosso, 2021, Dereziński et al., 2023, Heuveline et al., 2021).