# The New Integral Method for Analysing Function Series in Real and Complex Variables

Julio Cesar Jaramillo Quiceno

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Julio Cesar Jaramillo Quiceno. The New Integral Method for Analysing Function Series in Real and Complex Variables. 2025. hal-05143108v2

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# The New Integral Method for Analysing Function Series

## in Real and Complex Variables

Julio Cesar Jaramillo Quiceno

*Department of Physics, Faculty of Mathematics*

Universidad Nacional de Colombia

https://orcid.org/0000-0002-3518-668

jcjaramilloq@unal.edu.co

## Abstract

We introduce a new integral method to analyse the convergence of function series, particularly those that fail classical criteria such as the Weierstrass M-test. By transforming series into integrals evaluated at fixed points or along complex contours, we establish new convergence conditions based on inverse derivatives and exponential regularisation. The method applies to divergent power series, q-calculus, and special functions, and reveals uniform convergence even in oscillatory or singular settings. Applications include Green's functions in electrodynamics, propagators in quantum field theory, nucleon form factors, and neural networks with divergent loss functions. This unified framework bridges real and complex analysis, offering a robust tool for studying function series in pure mathematics, theoretical physics, and machine learning.

**Keywords:** series-to-integral; integral-to-series; special functions.

## 1 Introduction

Newton's 1665 observation that binomial expressions with fractional exponents can be expanded into infinite series [31] marks a foundational moment in the history of mathematical analysis, although it remained unpublished until the *Epistola Prior*. This insight laid the groundwork for subsequent advances by major figures in the field. Taylor [33] introduced generalised series expansions around arbitrary points in his *Methodum Incrementorum Directa et Inversa*, while Maclaurin [27] later specialised these expansions to the case around zero. Fourier's [11] seminal *Théorie Analytique de la Chaleur* introduced trigonometric series solutions to partial differential equations in physics, and Cauchy's [7] *Cours d'Analyse* provided a formal structure for convergence theory.

Subsequent refinements by Dini and Weierstrass culminated in the well-known M-test for uniform convergence [24]. Nonetheless, a wide class of function series fails to satisfy the M-test criterion, prompting the development of alternative approaches. In this work, we propose a new integral method grounded in a structural proposition that equates infinite
sums of infinite products of functions with corresponding function series. This idea draws
inspiration from classical complex analysis [6, 13] and the theory of divergent series [16].

In the real-variable setting, the method analyses convergence over open intervals (a, b) at selected evaluation points x₀ ∈ (a, b). In the complex case, we examine contour integrals over closed paths Γ that enclose candidate convergence points z₀, including singularities, thereby extending techniques from [14]. The resulting integrals generate infinite numerical series; however, even uniformly convergent function series may yield divergent numerical values at specific locations. These instances require careful evaluation using sequences designed to ensure convergence [26]. Constant function series are addressed using exponential approximations and auxiliary constructions [1].

The method generalises to power series [1, 12], integral representations of special functions [28], and *q*-calculus [18]. Its applicability extends to classical electrodynamics [19, 20], quantum field theory [32], nucleon form factors [15], and quantum statistical mechanics [25]. Convergence points are classified either as numerical sequences in Weierstrass-convergent settings [24], or as interior points of divergent intervals or singularities enclosed by contours [14, 35]. This unified framework synthesises classical methodologies [34, 6] and incorporates recent results on integral representations [30].

Finally, we present novel applications of the method to machine learning and artificial intelligence. It provides a robust regularisation technique for divergent series emerging in non-convex loss landscapes, and addresses convergence issues in deep learning models, including the neural tangent kernel (NTK) [38] and physics-informed neural networks (PINNs) [39, 40].

## Key Contributions

This work makes the following fundamental advances in the analysis of function series and their applications:

First, we establish a novel convergence criterion that successfully addresses the well-documented limitations of the Weierstrass *M*-test for oscillatory series, particularly evident in cases such as $\sum \sin(n^2x)/n$, with rigorous demonstration provided in Figure 1. The method introduces integral transformations and exponential regularisation techniques that stabilise divergent behaviour while preserving analytic structure.

Second, we present a methodological unification that seamlessly integrates real and complex analysis through a system of interconnected integral representations (Eqs. (3), (9), (18), creating a unified framework for series analysis. This includes generalisations to contour integration for singularities and numerical sequences, with new criteria for uniform convergence in both real and complex domains.

Third, we develop extensions to power series and *q*-calculus, deriving integral representations for divergent power series using auxiliary functions $h_n(x_0)$ and adapting the method to *q*-integrals and *q*-factorials. These provide new tools for analysing series that escape classical convergence tests.

Fourth, the work demonstrates significant applications in theoretical physics, including: the construction of Green's functions for non-dispersive media in classical electrodynamics; regularisation of propagators in quantum field theory without *iϵ* prescriptions; discrete representations of nucleon form factors; and novel approaches to occupation probabilities in quantum statistical mechanics.

Fifth, we introduce regularisation techniques for machine learning, addressing divergent series in non-convex loss landscapes, stabilising the Neural Tangent Kernel in infinite-
width networks, and resolving convergence issues in Physics-Informed Neural Networks through contour integration methods.

Sixth, the framework yields new integral representations for special functions, including novel series expansions for the Gamma function, Riemann zeta function, Laguerre and Hermite polynomials, Jonquière's function, and Bernoulli numbers, alongside contour-integral formulations for Dirichlet series.

Seventh, the work advances complex analysis by enabling evaluation of contour integrals at non-residue points and providing classification schemes for convergence points in complex series. This includes handling branch points and numerical sequences where traditional residue theory fails.

Eighth, the research bridges pure mathematics with theoretical physics and machine learning, demonstrating how abstract series analysis can solve concrete problems in quantum field theory, electrodynamics, and deep learning optimisation. The interdisciplinary synthesis offers practical tools for nonlinear PDEs and neural network training.

Additional innovations include: auxiliary exponential techniques for handling null derivatives; structural results connecting infinite sums of products to simple series; and a unified schematic for regularising divergent series through integral transformations. The combined theoretical and applied contributions position this work as a significant advancement in mathematical analysis and its modern applications.

## 2 Proposition

Let $f_n$ denote a function defined on a real or complex domain. We propose the following formal identity:

$$ \sum_{k=0}^{\infty} \prod_{n=1}^{\infty} f_{n,k} = \sum_{n=1}^{\infty} f_n. \qquad (1) $$

This expression suggests that, under suitable convergence conditions, an infinite sum over infinite products of doubly indexed functions $f_{n,k}$ reduces to a single function series $\sum f_n$. To motivate equation (1), we consider its finite analogue:

$$ \sum_{k=0}^{m} \prod_{n=1}^{p} f_{k,n} = \prod_{n=1}^{p} \left( 1 + \sum_{j=0}^{m-1} \frac{f_{n,j}}{f_{n,m}} \right) f_{n,m}. \qquad (2) $$

Taking the limits $m, p \to \infty$ and applying standard results concerning the interchange of summation and product for absolutely convergent series and rearranged infinite products (see [2]), we formally recover:

$$ \sum_{n=1}^{\infty} \sum_{j=0}^{\infty} f_{j,n} = \sum_{n=1}^{\infty} f_n. $$

Thus, equation (1) is justified in analytic contexts where such manipulations are permissible. The convergence behaviour of (1) must be analysed with respect to the nature of the variable domain. In the real setting, we consider open intervals $(a, b) \subset \mathbb{R}$, whereas in the complex setting, we work with closed contours $\Gamma \subset \mathbb{C}$ enclosing candidate convergence points $z_0$. In either domain, the method proceeds by evaluating each $f_n$ through integration and by ensuring the appropriate structural properties of the composite series. These considerations underpin the analytic regularisation techniques developed in the subsequent sections.
### 3 New Integral Method: Real Case

Consider a series of functions defined on a real interval $(1, \infty)$. Applying equation (1), we substitute $x = x_k$, multiply by $\Delta x_k$, and invoke the infinite product structure as discussed in [2]. Approximating $f_{n,k}(x_k) - 1 \approx -e^{-f_{n,k}(x_k)}$, we integrate at a reference point $x_0 \in (1, \infty)$ to obtain:

$$ \int_{1}^{\infty} f_n(x) dx = \sum_{n=1}^{\infty} \frac{1}{f'_{n}(x_0)}, \qquad (3) $$

where $f'_{n}(x_0) \neq 0$. This identity provides an integral representation for function series divergent under classical criteria. Notably, if the series is uniformly convergent according to the Weierstrass M-test, equation (3) does not hold, but it remains valid for divergent series over $(1, \infty)$.

For a general interval $(a, b)$, the method yields:

$$ \int_{a}^{b} f_n(x) dx = \sum_{n=0}^{\infty} \frac{1}{f'_{n}(x_0)}, \quad x_0 \in (a, b), \quad f'_{n}(x_0) \neq 0. \qquad (4) $$

In the uniformly convergent case, points $x_0 \in (a, b)$ are interpreted as numerical sequences, following [24], leading to:

$$ \int_{a}^{b} f_n(x) dx = \sum_{n=0}^{\infty} \frac{1}{f'_{n}(x_0)^{(n)}}, \quad \text{with } f'_{n}(x_0)^{(n)} \neq 0. $$

In scenarios where $f'_{n}(x_0) = 0$, such as constant function series, we approximate:

$$ e^{-f_n(x_0)} \approx 1 - f'_{n}(x_0), $$

which, when substituted into (3), leads to the exponential representation:

$$ \int_{a}^{b} f_{n}(x) dx = \sum_{n=0}^{\infty} e^{f_{n}(x_{0})}. \qquad (5) $$

For instance, using the auxiliary function $f_n(x_0) = -nf(x_0) + \ln(-1)^n$, we estimate:

$$ \int_{a}^{b} f_{n}(x) dx \approx \frac{1}{1 + e^{-f(x_{0})}}, $$

where divergence is avoided by applying the substitution $-f_n(x_0)$ when $f_n(x_0) > 0$.

Generalising from [5, 34], we propose a modified uniform convergence criterion:

$$ \int_{a}^{b} \|f\| dx = \sum_{n=0}^{\infty} \left\| \frac{1}{f'_{n}(x_0)} \right\| - \int_{a}^{b} \|f_n - f\| dx, \qquad (6) $$

where $f(x) = \sum_{n=0}^{m-1} f_n(x)$, and $f'_{n}(x_0) \neq 0$. By Cauchy's criterion, we obtain:

$$ \int_{a}^{b} \|f\| dx = \int_{a}^{b} \left\| \sum_{n=0}^{\infty} f_{n}(x) \right\| dx = \sum_{n=0}^{\infty} \left\| \frac{1}{f'_{n}(x_0)} \right\|. \qquad (7) $$
If $f'_n(x_0) = 0$, the term $\sum \|1/f'_n(x_0)\|$ in (7) is replaced by the exponential series from (5).

Let $x_0 \in (0, \pi)$ be fixed. Consider the sequence $f_n(x_0) = \frac{\sin(n^2x_0)}{n}$, which oscillates rapidly as $n \to \infty$ while its amplitude decays as $1/n$. We observe:

$$|f_n(x_0)| \le \frac{1}{n},$$

yet the harmonic series diverges, and hence the classical Weierstrass M-test fails. The functional series $\sum f_n(x)$ is not uniformly convergent in this case.

In contrast, we now apply the exponential criterion:

$$e^{f_n(x_0)} = 1 + f_n(x_0) + \frac{f_n(x_0)^2}{2!} + \dots$$

Since $f_n(x_0) \to 0$, it follows that $e^{f_n(x_0)} \to 1$. Moreover, the deviation $e^{f_n(x_0)} - 1$ decays faster than $1/n$ for most values of $x_0$, and the series $\sum (e^{f_n(x_0)} - 1)$ may converge conditionally due to oscillatory cancellation. Hence, the exponential sum

$$\sum_{n=1}^{\infty} e^{f_n(x_0)}$$

is significantly better behaved than the original series $\sum f_n(x_0)$, and may in fact converge, revealing behaviour not captured by classical tests.

Figure 1 displays the comparison between the function $f_n(x_0)$, the classical Weierstrass bound $M_n = 1/n$, and the exponential transformation $e^{f_n(x_0)}$, evaluated at $x_0 = \pi/4$.
<figure><img src="image_2.png" /><figcaption>Figure 1: Visual comparison of the function $f_n(x_0) = \frac{\sin(n^2 x_0)}{n}$, the classical bound $M_n = 1/n$, and the exponential transformation $e^{f_n(x_0)}$ at $x_0 = \pi/4$. While the classical bound diverges, the exponential criterion provides a bounded and stable comparison, demonstrating improved convergence behaviour.</figcaption></figure>

This example highlights the limitations of the Weierstrass test in treating highly oscillatory function sequences, and showcases the potential of exponential techniques for detecting convergence under milder analytical conditions. Such methods may prove particularly useful in Fourier and quantum analysis, where oscillatory decay and non-uniform convergence are commonplace.

# 4 New Integral Method: Complex Case

We extend the framework of equation (1) to complex variables. Specifically, if a complex-valued function series is uniformly convergent, then the infinite sum over infinite products also converges uniformly within a closed contour $\Gamma \subset \mathbb{C}$. This leads to the complex formulation of our integral method:

$$ \oint_{\Gamma} \sum_{n=1}^{\infty} f_n(z) dz = \frac{2\pi i}{e} \sum_{n=1}^{\infty} \frac{1}{f'_n(z_0)}, \qquad (8) $$

where $f'_n(z_0) \neq 0$. For notational simplicity, we may equivalently write:

$$ \oint_{\Gamma} f_n(z) dz = \frac{2\pi i}{e} \sum_{n=0}^{\infty} \frac{1}{f'_n(z_0)}. $$
By analogy with the real-variable case, we define uniform convergence for complex function series at a point $z_0 \in \Gamma$ when the contour integral of $f_n(z)$ produces a convergent numerical series under standard convergence tests. The point $z_0$ may coincide with a singularity of $f_n(z)$, and the series must satisfy two conditions: (i) uniform convergence within the closed contour $\Gamma$, and (ii) containment within the convergence disc $\|z-z_0\| < r$.

This method particularly addresses function series that fail the Weierstrass M-test but remain convergent when integrated over suitable contours. For Weierstrass-convergent series, we modify equation (8) by interpreting the convergence point $z_0$ as a numerical sequence $\{z_0\}_{n=0}^{\infty}$, similar to the approach used in the real case. In this setting, we consider:

$$f(z) = \sum_{n=0}^{\infty} f_n(z), \quad \text{and} \quad \sum_{n=0}^{\infty} \frac{1}{f'_n(z_0)} = \frac{1}{f'(z_0)}.$$

This yields a modified representation:

$$\oint_{\Gamma} f(z) dz = \frac{2\pi i}{e} \cdot \frac{1}{f'\{z_0\}_{n=0}^{\infty}}. \qquad (9)$$

Alternatively, the method may be expressed as:

$$\oint_{\Gamma} f(z) dz = \frac{2\pi i}{e} \sum_{n=0}^{\infty} \frac{1}{f'_{n}\{z_0\}_{n=0}^{\infty}}, \qquad (10)$$

assuming $f'_n(\{z_0\}) \neq 0$ for all $n$.

When $f'_n(z_0) = 0$, we propose a substitute involving a complex exponential series:

$$\oint_{\Gamma} f_n(z) dz = \frac{2\pi i}{e} \sum_{n=0}^{\infty} e^{f_n(z_0)}. \qquad (11)$$

To evaluate this expression numerically, we introduce an auxiliary function:

$$f_n(z_0) = -nf(z_0) + \frac{\ln(-1)^n}{i},$$

yielding:

$$\oint_{\Gamma} f_n(z) dz = \frac{2\pi}{e} \cdot \frac{1}{e^{-f(z_0)} + 1}, \qquad (12)$$

provided $f(z_0) \neq i\pi(2m-1)$, for $m \in \mathbb{N}$. Alternatively, using the auxiliary form $f_n(z_0) = nf(z_0)$, we obtain:

$$\oint_{\Gamma} f_n(z) dz = \frac{2\pi}{e} \cdot \frac{1}{1 - e^{f(z_0)}}, \qquad (13)$$

with the constraint $f(z_0) \neq 2im\pi$. In all cases, the total series $f(z_0) = \sum_{n=0}^{\infty} f_n(z_0)$ must be well-defined.

Uniform convergence along the contour $\Gamma$ additionally requires that $z_0$ lies strictly inside $\Gamma$, and that the integrand is analytic on and within $\Gamma$, so that Morera's theorem [28, 34] may be applied. Hence, for analytic $f_n$, the uniform convergence criterion becomes:

$$\oint_{\Gamma} \|f_n - f\| dz = \frac{2\pi i}{e} \sum_{n=0}^{\infty} \left\| \frac{1}{f'_{n}(z_0)} \right\|. \qquad (14)$$
If $f'_n(z_0) = 0$, we revert to the exponential formulation:

$$ \oint_{\Gamma} \|f_n - f\| dz = \frac{2\pi i}{e} \sum_{n=0}^{\infty} \|e^{f_n(z_0)}\|. $$

This provides an alternative criterion for uniform convergence of analytic function series in the complex domain.

We consider the sequence $f_n(z) = \frac{1}{(z-1/n)^{1.5}}$. Though the Weierstrass test fails, our method evaluates the inverse derivative:

$$ \frac{1}{f'_n(z_0)} = -\frac{1}{p}(z_0 - z_n)^{p+1}, \quad z_0 = i. \qquad (15) $$

This series exhibits rapid convergence as $|z_n - z_0|$ remains bounded away from zero.

<figure><img src="image_3.png" /><figcaption>Figure 2: Top: The complex contour $\Gamma$, the poles $z_n = 1/n$, and the evaluation point $z_0 = i$. Bottom: The convergence of the approximated integral $\oint f_n(z) dz$ via the derivative-inverse method, showing stable behaviour in both real and imaginary components.</figcaption></figure>

The proposed method offers a viable and powerful criterion for analysing the convergence of complex function series where classical techniques fall short. The use of inverse derivatives evaluated at a strategically chosen point not only captures the analytic structure but also ensures convergence of associated integrals within bounded contours.

# 5 Application to Power Series

Building on the contour integral approach, we now extend the method to analyse real power series and their convergence properties via auxiliary functions. In particular, we
derive an integral representation that yields the principal value of divergent series, inspired by results in [35].

<figure><img src="image_4.png" /><figcaption>Figure 3: A schematic overview of the integral method for regularising divergent series. The procedure applies real or complex integral transformations, followed by exponential regularisation when derivatives vanish. The result is a convergent formulation encompassing special functions and series, as developed in Sections 3–9.</figcaption></figure>

The diagram summarises a unified method for regularising divergent series via integral transformations. Starting from a general input real or complex the approach applies either definite or contour integration, yielding inverse derivative expressions. To ensure analytic continuity where derivatives vanish, an exponential regularisation step is introduced. The result is a convergent formulation encompassing special functions and physically meaningful series. This framework connects tools from functional and complex analysis, offering a robust criterion beyond classical convergence tests.

## 5.1 Proposition

Let $\sum_{n=0}^{\infty} a_n(x-\alpha)^n$ be a real power series centred at $\alpha$, with radius of convergence $R > 0$. The series converges uniformly in a neighbourhood of $x_0 \in (\alpha - R, \alpha + R)$, with $x_0 \neq \alpha$, if the auxiliary series

$$ \sum_{n=0}^{\infty} \frac{1}{h_n(x_0)} (x_0 - \alpha)^{n+1}, \quad \text{where } a_n = \frac{1}{h_n(x_0)}, \qquad (16) $$

is convergent, and $h_n(x_0)$ is divergent in the classical sense.
## 5.2 Integral Representation

Applying equation (16), we derive the following integral form for the associated power series:

$$ \int_a^b \frac{h_n(x)}{(x-\alpha)^n} dx = \sum_{n=0}^{\infty} \frac{1}{h_n(x_0)} (x_0 - \alpha)^{n+1}, \quad (17) $$

for a suitable interval $(a, b) \subset (\alpha - R, \alpha + R)$ containing $x_0$.

## 5.3 Proof

We begin by defining the differential quotient:

$$ f'_{n}(x_{0}) = \frac{f_{n}(x_{0}) - f_{n}(\alpha)}{x_{0} - \alpha}, $$

which, when substituted into the real-variable integral representation (cf. equation (17)), leads to:

$$ \int_{x_0-R}^{x_0+R} f_n(x) dx = \sum_{n=0}^{\infty} \left| \frac{x_0 - \alpha}{f_n(x_0) - f_n(\alpha)} \right| \le \sum_{n=0}^{\infty} \frac{|x_0 - \alpha|}{f_n(x_0)}. \quad (18) $$

Now let us assume a specific functional form:

$$ f_n(x) = \frac{h_n(x)}{(x - \alpha)^n}. $$

Substituting this into (18), we obtain:

$$ \int_{x_0-R}^{x_0+R} \frac{h_n(x)}{(x-\alpha)^n} dx = \sum_{n=0}^{\infty} \frac{|x_0 - \alpha|}{h_n(x_0)} (x_0 - \alpha)^n. \quad (19) $$

Therefore, even if the original coefficients $a_n$ diverge, the series can exhibit uniform convergence provided the auxiliary functions $h_n(x_0)$ satisfy the above condition.

This framework generalises classical results on term-wise integration and convergence [5, 24], and introduces a new criterion for uniform convergence at points $x_0 \ne \alpha$ within the domain of convergence of the power series. Let us consider the respective analysis for the function $h_n(x) = \frac{\sin(n^2x)}{n}$. The Figure 4 compares three distinct sequences evaluated at the point $x_0 = \pi/4$. The oscillatory function $h_n(x_0) = \sin(n^2x_0)/n$ is shown in blue, displaying characteristic amplitude decay proportional to $1/n$ while failing the Weierstrass M-test due to non-uniform convergence. The classical Weierstrass bound $M_n = 1/n$ appears as a black dashed line, serving as a divergent majorant series. In red, the exponential transformation $e^{h_n(x_0)}$ demonstrates superior convergence behavior, stabilizing the oscillations and approaching unity as $n$ increases. This visual comparison underscores the limitations of traditional convergence tests when applied to highly oscillatory functions, while highlighting the efficacy of the proposed exponential criterion in capturing convergence that eludes classical methods, particularly in contexts such as Fourier analysis and quantum mechanical systems where rapid oscillations are prevalent (see Section 3 for analytical details. The Figure 5 presents a comparative analysis between a divergent power series and its auxiliary convergent counterpart. The upper portion illustrates the partial sums of the divergent series $\sum a_n(x_0 - \alpha)^n$, where the coefficients $a_n = 1/h_n(x_0)$ induce unbounded growth, as evidenced by the orange trajectory. In contrast, the lower
<figure><img src="image_5.png" /><figcaption>Figure 4: The function $h_n(x_0) = \frac{\sin(n^2 x_0)}{n}$ (red) is bounded but fails the classical Weierstrass M-test (black), as the bound $M_n = \frac{1}{n}$ yields a divergent majorant. The exponential transformation $e^{h_n(x_0)}$ (blue) stabilises the oscillations, indicating improved convergence behaviour</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: Comparison between the original divergent power series $\sum a_n(x_0 - \alpha)^n$ (orange), with $a_n = 1/h_n(x_0)$, and the proposed auxiliary series $\sum \frac{1}{h_n(x_0)}(x_0 - \alpha)^{n+1}$ (green). While the classical expansion diverges, the auxiliary series converges uniformly for $x_0 \in (\alpha - R, \alpha + R)$, illustrating the power of the new integral method.</figcaption></figure>

portion displays the behavior of the auxiliary series $\sum (x_0 - \alpha)^{n+1}/h_n(x_0)$ in green, which maintains uniform convergence within the interval $x_0 \in (\alpha - R, \alpha + R)$. This striking divergence-convergence dichotomy validates the integral method's capacity to regularize problematic series through careful contour selection and auxiliary function construction, as developed in Section 5. The graphical representation reinforces the theoretical framework by demonstrating how strategic transformations can extract meaningful convergence from otherwise intractable series expansions.

# 6 Extension to Complex Power Series

The proposed methodology naturally extends to complex power series via contour integration techniques. Let $\sum_{n=0}^{\infty} a_n(z - \alpha)^n$ be a complex power series with radius of convergence $R > 0$. We define an associated integral representation of the form:

$$ \oint_{\Gamma} \frac{h_n(z)}{(z - \alpha)^n} dz = \sum_{n=0}^{\infty} \frac{1}{h_n(z_0)} (z_0 - \alpha)^{n+1}, \qquad (20) $$

where $\Gamma$ is a closed contour entirely contained within the convergence disc $|z - \alpha| < R$, and $z_0 \in \Gamma$, with $z_0 \neq \alpha$, is a point selected to evaluate convergence. We now state a convergence criterion for uniform convergence of complex power series.
## Proposition

Let $\sum_{n=0}^{\infty} b_n(z - \beta)^n$ be a complex power series. The series converges uniformly if the auxiliary series

$$ \sum_{n=0}^{\infty} \frac{|z_0 - \beta|}{h_n(z_0)} (z_0 - \beta)^n \quad (21) $$

converges, where $b_n = \frac{1}{h_n(z_0)}$, and the point $z_0$ satisfies:

(i) $z_0 \in \Gamma$,

(ii) $z_0 \neq \beta$,

(iii) $h_n(z_0)$ is divergent in the classical sense.

Analogous to the real-variable case, we derive an integral representation from (20):

$$ \oint_{\Gamma} \frac{h_n(z)}{(z - \beta)^n} dz = \frac{2\pi i}{e} \sum_{n=0}^{\infty} \frac{|z_0 - \beta|}{h_n(z_0)} (z_0 - \beta)^n, \quad (22) $$

with $\Gamma \subseteq \{z \in \mathbb{C} \mid |z - \beta| < R\}$. The constraint $h_n(z_0) \neq 0$ must hold for all $n$.

In the limiting case where $h_n(z_0) = 0$, we propose a reinterpretation via substitutions:

$$ \frac{|x_0 - \alpha|}{h_n(x_0)} \rightarrow h_n(\alpha), \quad \frac{|z_0 - \beta|}{h_n(z_0)} \rightarrow h_n(\beta), $$

which leads to the following reformulated integral representations:

$$ \int_a^b \frac{h_n(x)}{(x - \alpha)^n} dx = \sum_{n=0}^{\infty} h_n(\alpha)(x_0 - \alpha)^n, \quad (23) $$

$$ \oint_{\Gamma} \frac{h_n(z)}{(z - \beta)^n} dz = \frac{2\pi i}{e} \sum_{n=0}^{\infty} h_n(\beta)(z_0 - \beta)^n. \quad (24) $$

These integral representations provide an alternative characterisation of power series expansions, bypassing standard differentiation techniques. Related approaches have been explored previously using Green's functions and polylogarithmic representations [30], as well as purely algebraic constructions without reliance on Cauchy's integral formula [26].

# 7 Application to Special Functions

Many classical special functions admit both integral and infinite series representations, typically valid within certain regions of convergence. Using our integral method, we derive alternative series expansions from integral definitions and, conversely, obtain contour integral representations from known series expressions.

We classify the results as follows:

* Integral representations → Series expansions

* Series representations → Contour integral formulations
# 7.1 From Integral to Series Representations

The series expansions derived via our method depend critically on the choice of evaluation point $x_0$ within the domain of integration.

## Gamma Function

Based on classical results in [34], we apply the real-variable integral method (cf. equation (17)) to obtain a novel expansion of the Gamma function for positive real arguments:

$$ \Gamma(t_0) = \sum_{x=0}^{\infty} \frac{t_0^{2-x} e^{t_0}}{x - t_0 - 1}, \quad t_0 \in (0, \infty). \qquad (25) $$

## Riemann Zeta Function

In combination with the identity linking $\zeta(s)$ and $\Gamma(s)$, and following approaches in [1, 6], we obtain:

$$ \zeta(x_0) \Gamma(x_0) = \sum_{s=0}^{\infty} \frac{x_0^{1-s} (e^{x_0} - 1)^2}{s(e^{x_0} - 1) - x_0 e^{x_0}}, \quad x_0 \neq 0, \quad \Re(s) > 0. \qquad (26) $$

## Exponential Integral

Adapting the formulation from [1], we derive a representation of the exponential integral function $E_n(x)$ centred at a point $u_0$ as:

$$ E_{u_0}(x) = \sum_{n=0}^{\infty} \frac{u_0^{n+1} e^{u_0 x}}{x u_0 + n}, \quad x > 0. \qquad (27) $$

This expansion is valid for fixed $u_0 \in \mathbb{R}^+$ and extends classical definitions based on asymptotic or series expansions of exponential integrals.

# 7.2 From Contour Integrals to Infinite Series

We now derive series representations from contour integrals evaluated at points $z_0$, which may include isolated singularities located within the integration path. These expansions offer alternative formulations of classical special functions.

## Pochhammer Integral

Extending the framework of the Beta function as developed in [6], we obtain a series approximation for the Pochhammer integral:

$$ B(t_0) \approx 2\pi \sum_{p,q=2}^{\infty} \frac{i t_0^{2-p} (1-t_0)^{2-q} \cdot 4 \sin(\pi p) \sin(\pi q) e^{i\pi(p+q)-1}}{(p-1) t_0^{p-2} (1-t_0)^{q-1} - t_0^{p-1} (1-t_0)^{q-2}}, \qquad (28) $$

valid for $t_0 \in \mathbb{N}$, $t_0 \neq 1$, and with convergence determined by the contours specified in [6].

## Schläfli Integral

Synthesising results from [2, 6, 12], we derive the following expansion:

$$ P_{t_0}(z) = \frac{\pi^2}{e} \sum_{n=0}^{\infty} \frac{2^{n+2}(t_0 - z)^{n+2}}{(t_0^2 - 1)^n} \cdot \frac{(t_0^2 - 1)}{[2t_0n - (n+1)(t_0^2 - 1)]}. \qquad (29) $$
## Laguerre Polynomials

For the Laguerre polynomials, following [2], the method yields:

$$L_{t_0}(z) = 4i\pi^2 \sum_{n=0}^{\infty} \frac{(1-t_0)^4 t_0^{n+1} e^{\frac{zt_0}{1-t_0}-1}}{1-(n+1)(1-t_0)^3+(1-t_0)^2}, \quad t_0 \in \mathbb{N}. \quad (30)$$

## Hermite Polynomials

Following the formulation in [2], we obtain a series expression for the Hermite polynomials:

$$H_{t_0}(z) = 4i\pi^2 \sum_{n=0}^{\infty} \frac{e^{(t_0-z)^2 - (z^2+1)t_0^{n+1}}}{n! [2(z-t_0) - (n+1)t_0^{-1}]}, \quad t_0 \in \mathbb{N}. \quad (31)$$

## Jonquière's Function

Developing the approach in [21], we obtain a series expansion for Jonquière's function, which is classically defined via the polylogarithm:

$$\sum_{n=1}^{\infty} \frac{x^n}{n^s} = \frac{2\pi i}{\Gamma(s)} \sum_{n=1}^{\infty} \frac{z_0^{1-s} x^{-n} e^{nz_0}}{(s-1)z_0^{-1} - n}, \quad (32)$$

valid for $\Re(s) > 1$, with $x \in \mathbb{C}$ and $z_0$ chosen within an appropriate integration contour.

## Complex Gamma Function

From the works [1, 12, 13], we derive two series expansions for the complex Gamma function. First, an approximation valid for $\Re(z_0) > 0$:

$$\Gamma(z_0) \approx \sum_{n=0}^{\infty} \frac{z_0^{2-n} e^{z_0}}{n - z_0 - 1}. \quad (33)$$

Second, a full expansion involving reflection and sine terms:

$$\Gamma(z) = \frac{\pi}{e \sin(\pi z)} \sum_{n=0}^{\infty} \frac{(nz)^{z+2} e^{nz}}{(n-1)z+1}, \quad |z| < \infty. \quad (34)$$

## Bernoulli Numbers

Extending the integral representation framework from [2], we obtain a series expression for the Bernoulli numbers:

$$B_{z_0} = \frac{2\pi i}{e} \sum_{n=0}^{\infty} \frac{(e^{z_0} - 1)^2 z_0^{n+1}}{(1 - e^{z_0})n - z_0 e^{z_0}}, \quad z_0 \in \mathbb{N}, \quad (35)$$

where the integration contour encloses the origin in the complex plane (counter-clockwise), with $|z| < 2\pi$, and excludes poles at $z = 2\pi in, n \in \mathbb{Z} \setminus \{0\}$.
## 7.3 From Infinite Series to Contour Integral Representations

Many transcendental functions defined via Taylor or Dirichlet series admit equivalent contour integral representations when expressed through the integral formalism introduced in Section 4. We illustrate this correspondence with several examples.

### (i) The Dilogarithm Function

The dilogarithm function $Li_2(z)$, as defined by Maximon [29], admits the series representation:

$$Li_2(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^2},$$

valid in the domain $|z| \le 1$. Applying equation (24), we obtain the following contour integral representation:

$$Li_2(n) = \frac{e}{2\pi i} \oint_{\Gamma} \frac{n^2 z^{1-n}}{1-n} dz, \quad n \neq 1, \qquad (36)$$

where $\Gamma$ is a closed contour enclosing the origin.

### (ii) The Binomial Series

The binomial expansion, as presented by Markushevich [28], may also be expressed in contour integral form. Applying equation (8) and replacing $z_0$ with $z$, we obtain:

$$(1+z)^{\alpha} \approx \frac{ie}{2\pi} \oint_{\Gamma} \frac{n!(\alpha-n)!}{\alpha!(n-1)} z^{1-n} dz, \qquad (37)$$

valid for $\alpha \in \mathbb{C}$, $n \neq 1$, and $|z| < 1$.

### (iii) The Lerch Functions

The Lerch transcendent, defined in [29, 12], admits the series expansion for $|z| < 1$, and the corresponding contour integral representation is given by:

$$\Phi(m, s, \alpha) = \frac{e}{2\pi i} \oint_{\Gamma} (m + \alpha)^s \cdot \frac{z^{1-m}}{1-m} dz, \quad m \neq 1. \qquad (38)$$

Berndt [3] proposed a specific integration path for Lerch-type integrals: a positively oriented semicircular contour centered at $(c, 0)$ with radius $m + \frac{1}{2} - c$, where $c \in \mathbb{R}$. Adopting this contour, the Lerch function admits the alternative contour integral representation:

$$\varphi(u, a, s) = \oint_{\Gamma} \frac{i(z+a)^s}{2\pi z} e^{-2\pi izu} dz. \qquad (39)$$

Further details on complex representations of Lerch functions and related polylogarithmic functions can be found in [3]. These examples illustrate that transcendental functions initially expressed via power series can equivalently be formulated as contour integrals through the application of equation (24).
(iv) *The Riemann Zeta Function*

A contour integral representation for the Riemann zeta function is given by:

$$ \zeta(n) \approx \frac{e}{2\pi i} \oint_{\Gamma} \frac{n^z}{\ln n} dz, \qquad (40) $$

where $\Gamma$ is a vertical contour in the complex plane defined by $\Re(z) > 0$, with $n > 0$, as presented in [12].

(v) *Dirichlet Series*

Dirichlet series generalise the Riemann zeta function and are foundational in analytic number theory [16]. A contour integral formulation of a general Dirichlet-type function is given by:

$$ f(a_n) = \frac{1}{2\pi i} \oint_{\Gamma} \frac{1}{a_n \lambda_n} e^{\lambda_n z + 1} dz, \qquad (41) $$

where $\{a_n\} \subset \mathbb{C}$ is a complex sequence, and $\{\lambda_n\}$ is a decreasing, convergent real sequence. The integral is taken over a closed contour $\Gamma$ in the complex plane. To ensure convergence of (41), it is required that $a_n \gg \lambda_n$, i.e., the growth of $a_n$ dominates that of $\lambda_n$.

# 8 Applications to q-Calculus

The theory of q-derivatives was introduced by Jackson in 1908 [18] as a discrete analogue of ordinary derivatives. For a function $f_n(x)$, the q-derivative is defined by:

$$ D_q f(x) = \frac{f(qx) - f(x)}{(q-1)x}, \qquad (42) $$

where $q \in \mathbb{R} \setminus \{1\}$ and $x \neq 0$. This difference operator recovers the standard derivative in the limit $q \to 1$.

To construct an analogue of the integral method defined in equation (17), we evaluate the q-integral over an interval $[a, b] \subset \mathbb{R}$, centred at a point $x_0 \in [a, b]$, yielding:

$$ \int_a^b f_n(x) \, d_q x = \sum_{n=0}^{\infty} \left| \frac{1}{D_q f_n(x_0)} \right| \qquad (43) $$

Substituting (42) into (43), we obtain:

$$ \sum_{n=0}^{\infty} \left| \frac{(q-1)x_0}{f_n(qx_0) - f_n(x_0)} \right| \le \sum_{n=0}^{\infty} \left| \frac{(q-1)x_0}{f_n(x_0)} \right| \qquad (44) $$

Now, consider the particular form $f_n(x_0) = \frac{a_n}{[x-x_0]_n}$, where $[x-x_0]_n$ denotes the q-analogue of the falling factorial:

$$ [x - x_0]_n = (x - x_0)(x - qx_0)(x - q^2 x_0) \cdots (x - q^{n-1} x_0), \quad [x - x_0]_0 = 1. $$

Substituting into (44), we obtain the following q-integral representation:

$$ \int_a^b \frac{a_n \, d_q x}{[x - x_0]_n} = (1 - q)x_0 \sum_{n=0}^{\infty} d_n[x - x_0]_n, \qquad (45) $$
where $d_n = \frac{1}{a_n}$. This formulation provides a q-analogue of the power series integral method within a finite convergence interval $[a, b]$.

In the complex domain, the method generalises via contour integration. Let $\Gamma$ be a closed contour enclosing $z_0 \in \mathbb{C}$. Then:

$$ \oint_{\Gamma} \frac{a_n \, d_q z}{[z - z_0]_n} = (1 - q) z_0 \sum_{n=0}^{\infty} h_n[z - z_0]_n, \qquad (46) $$

where $h_n = \frac{1}{a_n}$ and $[z - z_0]_n$ denotes the complex q-factorial term as above. This result extends the integral method to q-calculus and defines convergence criteria for q-power series in both real and complex settings.

# 9 Applications in Physics and Discussion

The integral method developed in both real and complex domains, with particular emphasis on the complex case, finds applications in a range of physical and computational frameworks. Notably, it enables the evaluation of contour integrals at points where Cauchy's residue theorem fails, such as numerically defined sequences or branch points. The method is applied in classical electrodynamics, quantum field theory, form factor analysis in nuclear physics, quantum statistical mechanics, and optimization tasks in artificial intelligence.

## 9.1 Application to Classical Electrodynamics

Maxwell's equations reduce to wave equations for the electric and magnetic fields. These are solved using time-dependent Green's functions—specifically, the retarded and advanced forms—constructed via contour integration. Standard derivations rely on residues at poles $k = \omega/c$ on the real axis [19]. However, points such as $k = \pm i\omega/c$ are not poles, rendering the residue theorem inapplicable. Using equation (12), we construct Green's functions from points $ink_n c = \omega_n$ (retarded) and $-ink_n c = \omega_n$ (advanced), where $k_n = m\beta_n$ or $k_n = \ell\beta_n$ define discrete indices. This leads to complex-valued retarded/advanced fields in nondispersive media. When $\omega_n = ck_n$, the fields vanish. To obtain nonzero contributions, the relation $\omega_n = \pm nck_n$ must hold. The formulation extends to the covariant case, leading to discrete four-potentials [20].

## 9.2 Application to Quantum Field Theory

In scalar Yukawa theory for two distinguishable particles, scattering amplitudes are evaluated using Feynman rules [32]. By applying equation (12), discrete attractive and repulsive Yukawa potentials emerge when $q_0 = -inm_\phi$. Similarly, propagators for scalar, fermionic, and electromagnetic fields can be constructed using the integral method at points where standard contour methods break down. The continuous propagators are defined in [14, 32, 37], while discrete versions appear as Fourier series expansions. For example, scalar propagators with singularities at $p = \omega$ are regularised without the $i\epsilon$ prescription by using $p = n\omega_p$, $p = nm$, or $q_0 = \sqrt{-in\epsilon}$. Contour selection (upper/lower
semicircle) depends on time ordering $x^0 < y^0$ or $x^0 > y^0$ [32]. This approach recovers discrete Pauli–Jordan propagators with positive and negative frequency contributions from [14].

## 9.3 Application to Form Factors

Applications to nucleon structure involve four cases. First, form factors arising from inelastic scattering are represented as discrete charge densities using (12), evaluated on imaginary-axis contours determined by the particle mass. Second, for confined systems, mass-$m$ particles in bounded domains lead to discrete form factor series [17]. Third, equation (17) provides auxiliary function expansions of matrix elements consistent with quark distributions [15]. Fourth, the proton structure function $F_2$, evaluated for $x_0 \in [0, 1]$, admits a parton-distribution-based series representation via the real integral method [22].

## 9.4 Application to Quantum Statistical Mechanics

In the context of quantum gases, equation (1) determines the average number of bosons and fermions over energy levels $\epsilon_k$. This framework covers Bose–Fermi mixtures, such as ultracold $^6$Li–$^7$Li condensates [25], yielding discrete occupation probabilities without the need for zeta-regularisation or cutoff schemes.

## 9.5 Application to Artificial Intelligence

The integral method offers novel regularisation tools in machine learning and optimization. In training deep models, divergent series often arise in the expansion of non-convex loss functions. Equation (17) regularises these expansions. For a loss $\mathcal{L}(\theta)$, the Hessian expansion $H(\theta) = \sum_{n=0}^{\infty} \partial_{\theta}^{n} \mathcal{L}$ yields a regularised integral:

$$ \oint_{\Gamma} H(\theta) \, d\theta = \frac{2\pi i}{e} \sum_{n=0}^{\infty} \frac{1}{\partial_{\theta} H_n(z_0)}, \quad z_0 \in \Gamma. \qquad (47) $$

In infinite-width neural networks [38, 39], the Neural Tangent Kernel (NTK) involves function series with depth-induced divergences. Our method resolves these by controlling activation convergence and parameter growth. Moreover, in Physics-Informed Neural Networks (PINNs), loss divergence occurs when the Neumann series $\sum_{n=0}^{\infty} (\mathcal{N}')^n$ fails [40]. Applying the contour method with a Hankel path $\Gamma$, we define:

$$ \mathcal{L}_{\text{reg}} = \frac{2\pi i}{e} \oint_{\Gamma} \frac{\mathcal{N}[u_{\theta}](z)}{(z - x_n) \ln(1 + z)} dz, \qquad (48) $$

where the singularities $\{z_k\}_{k=1}^K$ are determined by:

$$ z_k = \lim_{n \to \infty} \frac{x_{n+1} - x_n}{x_n - x_{n-1}}. \qquad (49) $$

This construction yields stable PINN training dynamics in otherwise unstable regimes.
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