# Loci of exactness for the real and imaginary parts of truncated Taylor series of analytic functions

Gaëtan Hello

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Gaëtan Hello. Loci of exactness for the real and imaginary parts of truncated Taylor series of analytic functions. 2025. hal-05020392

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# Loci of exactness for the real and imaginary parts of truncated Taylor series of analytic functions

Gaëtan Hello

Université Paris-Saclay, Univ Evry, LMEE
91020, Evry, France

gaetan.hello@univ-evry.fr

https://orcid.org/0000-0002-3560-4707

March 17, 2025

## Abstract

Taylor series expansions of analytic functions are often truncated for practical uses. The overall accuracy of truncated series is generally believed to improve as the number of terms retained increases. However, the graphical representation of the real and imaginary parts of the difference between analytic functions and their truncated series exhibits loci where truncated series can provide exact values. In particular, near the expansion point, the loci of exactness are observed to gather on curves emanating from it. The number and tangential angles of these curves at the expansion point are shown to be related to the exponent and the coefficient of the first term in the remainder associated with the truncated series. These curves of loci of exactness can be further studied from a mathematical perspective and be employed, for instance, in the design of analytical and numerical procedures for the accurate identification of series coefficients.

**Keywords:** complex analysis, analytic functions, holomorphic functions, Taylor series, truncated series.

**MSC Classification:** 30B10.

## 1 Introduction

Taylor series are extensively used to represent or approximate functions in mathematics, physical sciences, and engineering. While the concept of this ubiquitous mathematical object was coined after Taylor's monograph of 1715 [1], earlier mentions can be found in Fifteenth-Century India by Madhava and successors, and in Seventeenth-Century Europe by J. Gregory in 1671 and I. Newton in 1691-92 [2, ch 1&12]. The extension of Taylor series to functions of a complex variable was largely consolidated by the end of the nineteenth century with notable contributions by A. L. Cauchy [3, ch
1; sec 1.5] [4, ch 6&10]. Beyond their intrinsic interest in mathematics, Taylor series of analytic functions have also proved their usefulness in the expression and resolution of numerous problems in science and engineering [5, 6, 7, 8].

In order to perform practical calculations, Taylor series are often truncated up to a certain number of terms. It is sometimes believed that the more terms are retained, the more accurate is the representation of the analytic function by the truncated series. This belief certainly originates from the comprehension of Taylor series for functions of a real variable but has to be nuanced regarding analytic functions. Indeed, it can be observed in applications in fracture mechanics [9] that the error fields for the real and imaginary parts of the difference between analytic functions and their truncated series approximations exhibit more complex patterns than just a radial increase of the error. There indeed exist loci of points arranged in remarkable motifs where truncated series have the same values for the real or imaginary parts as the original analytic function.

The present study intends to extend this observation to the truncated Taylor series of general analytic functions. First, the existence and spatial distribution of these loci of exactness are numerically observed for various elementary analytic functions. Then, inspired by the strikingly regular patterns obtained, some formulae about the properties of these curves are derived. Eventually, the validity of these formulae is assessed by comparing their predictions with actual numerical results.

## 2 Definitions

Given a function $f$ analytic throughout an open disk of radius $R \neq 0$ centered at $z = z_0$, the function can be represented in the open disk as the convergent power series:

$$f(z) = \sum_{k=1}^{\infty} a_k \cdot (z - z_0)^{\lambda_k}, \text{ with } a_k \neq 0 \text{ and } \lambda_{k+1} > \lambda_k \quad (1)$$

At the difference with the usual expression for power series where the summation is performed with exponents in $\mathbb{N}$ and terms possibly equal to zero, Eq. 1 here only retains non-zero terms. This choice is motivated by the need to have access to the first non-zero term of the remainder in subsequent manipulations. The set of exponents $\{\lambda_k\}_{k \in \mathbb{N}^*} \subseteq \mathbb{N}$ then depends on the function and the expansion point considered. For $z_0 = 0$, it can be for example $\mathbb{N}$ for $\exp(z)$, $\mathbb{N}^*$ for $\log(1+z)$ or $2\mathbb{N}$ for $\sin(z)/z$.

The series expansion can then be additively decomposed as the sum of the truncated series with $n$ non-zero terms $s_n$:

$$s_n(z) = \sum_{k=1}^{n} a_k \cdot (z - z_0)^{\lambda_k} \quad (2)$$

And the complementary remainder $r_n$:

$$r_n(z) = f(z) - s_n(z) = \sum_{k=n+1}^{\infty} a_k \cdot (z - z_0)^{\lambda_k} \quad (3)$$

Associated with the truncated series, it is possible to define the loci of exactness of $s_n$ for its real and imaginary parts respectively as:

$$L_n^{\text{Re}}(f) = \{z \in \mathbb{C} \mid \text{Re}(f(z)) = \text{Re}(s_n(z))\} = \{z \in \mathbb{C} \mid \text{Re}(r_n(z)) = 0\} \quad (4)$$

And:

$$L_n^{\text{Im}}(f) = \{z \in \mathbb{C} \mid \text{Im}(f(z)) = \text{Im}(s_n(z))\} = \{z \in \mathbb{C} \mid \text{Im}(r_n(z)) = 0\} \quad (5)$$
### 3 Numerical observation of the loci of exactness

The present numerical investigation aims at observing empirically the loci $L_n^{\text{Re}}$ and $L_n^{\text{Im}}$ for the following analytic functions expanded at different $z_0$:

$$ 
\begin{align} 
\exp(z) &= \sum_{z_0=1+i}^{\infty} \frac{\exp(1+i)}{k!} \cdot (z - (1+i))^k \nonumber \\ 
\log(1+z) &= \log(1+i) + \sum_{k=1}^{\infty} \frac{(-1)^{k-1}}{k \cdot (1+i)^k} \cdot (z-i)^k, \quad |z-i| < \sqrt{2} \tag{6} \\ 
\frac{\sin(z)}{z} &= \sum_{z_0=0}^{\infty} \frac{(-1)^k}{(2k+1)!} \cdot z^{2k} \nonumber 
\end{align} 
$$

This sample of analytic functions, even if it does not encompass all their variety, allows to investigate the existence and property of the loci of exactness for different expansion points, different sets of exponents, for finite and infinite radii of convergence and for functions with possibly a branch point ($z = -1$ for $\log(1+z)$) or a removable singularity ($\lim_{z \to 0} \sin(z)/z = 1$).

Two terms are retained in truncated series for the examples considered in this section. Then, in order to observe the spatial distribution of the loci of exactness, the real and imaginary parts of $f(z) - s_2(z)$ are represented in Fig. 1 for the three analytic functions considered. The loci of exactness correspond to the points located on the 0-isovalues of the different fields. All three functions exhibit such isovalues for both their real and imaginary parts. While these isovalues are not limited to the neighborhood of the expansion point for $\exp(z)$ and $\sin(z)/z$ (not displayed on the figures), a remarkable pattern actually emerges near the expansion point where loci of exactness appear to aggregate on curves emanating from it. These curves all correspond in the examples to a change of sign for the real or imaginary part. An alternative, and perhaps more immediately comprehensible, way of representing these loci consists to plot the $\log_{10}$ of the absolute value of $\text{Re}(f(z) - s_2(z))$ and $\text{Im}(f(z) - s_2(z))$. This representation is proposed in Fig. 2. The same locations as in Fig. 1 emerge as points where the absolute error is close to the numerical threshold of $10^{-16}$.

These curves of loci of exactness emanating from the expansion point display a certain regularity in their relative positions. Inspired by these remarkable arrangements, the next section then intends to derive analytically some intrinsic properties about these curves.

### 4 Properties of the loci of exactness near the expansion point

The sets $L_n^{\text{Re}}$ and $L_n^{\text{Im}}$ correspond respectively to the loci where the real and imaginary parts of $r_n$ are equal to 0. Then starting from:

$$ r_n(z) = 0 \tag{7} $$

It is then possible to isolate the first term in the remainder series:

$$ a_{n+1} \cdot (z - z_0)^{\lambda_{n+1}} = - \sum_{k=n+2}^{\infty} a_k \cdot (z - z_0)^{\lambda_k} \tag{8} $$
<figure><img src="image_2.png" /><figcaption>Figure 1: Fields for: (a) Re $(f(z) - s_2(z))$, (b) Im $(f(z) - s_2(z))$.</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2: Fields for : (a) $\log_{10}|\operatorname{Re}(f(z) - s_2(z))|$, (b) $\log_{10}|\operatorname{Im}(f(z) - s_2(z))|$.</figcaption></figure>
With series coefficients satisfying $a_k \neq 0$ and introducing the polar form $z = z_0 + re^{i\theta}$ at the expansion point, the previous equation becomes:

$$a_{n+1} \cdot e^{i\lambda_{n+1}\theta} = - \sum_{k=n+2}^{\infty} a_k \cdot e^{i\lambda_k\theta} \cdot r^{\lambda_k - \lambda_{n+1}} \quad (9)$$

With $\lambda_k - \lambda_{n+1} \ge 1$ for $k \ge n+2$, taking to the limit Eq. 9 when $r \to 0^+$ leads to:

$$a_{n+1} \cdot e^{i\lambda_{n+1}\theta} = 0 \quad (10)$$

While there is no angular solution that cancels simultaneously the real and imaginary parts of $a_{n+1} \cdot e^{i\lambda_{n+1}\theta}$, both can be individually canceled.

For the real part, the angular solution satisfies:

$$\mathrm{Re}(a_{n+1}) \cdot \cos(\lambda_{n+1}\theta) - \mathrm{Im}(a_{n+1}) \cdot \sin(\lambda_{n+1}\theta) = 0 \quad (11)$$

Assuming the polar decomposition $|a_{n+1}| \cdot e^{i \arg(a_{n+1})}$ for the coefficient $a_{n+1}$, the set of angles that cancels the real part of Eq. 10 is then expressed as:

$$\theta_{n,k}^{\mathrm{Re}} = \frac{1}{\lambda_{n+1}} \cdot \left( \frac{\pi}{2} + k \cdot \pi - \arg(a_{n+1}) \right) \quad (12)$$

Following a similar approach, the set of angles that cancels the imaginary part of Eq. 10 is given by:

$$\theta_{n,k}^{\mathrm{Im}} = \frac{1}{\lambda_{n+1}} \cdot (k \cdot \pi - \arg(a_{n+1})) \quad (13)$$

Using Eqs.12,13 it becomes possible to correlate the number of curves of exactness and their tangential angle at the expansion point with the exponent $\lambda_{n+1}$ and the coefficient $a_{n+1}$ of the first term in the associated remainder.

# 5 Numerical assessment of loci's properties

The validation of Eqs.12,13 here relies on their ability to correlate the properties of the curves of loci of exactness near the expansion point for truncated series with two terms $s_2(z)$ (previously studied) and three terms $s_3(z)$.

Considering for example $\exp(z)$, Eq.12 leads to 4 unique values in $[0, 2\pi)$ for the real part of $s_2(z)$ (((2k + 1)·π - 2)/4, 0 ≤ k ≤ 3) while Eq.13 leads to 4 unique values for its imaginary part ((k·π - 1)/2, 1 ≤ k ≤ 4). For $s_3(z)$, Eq.12 leads to 6 unique values in $[0, 2\pi)$ for the real part (((2k + 1)·π - 2)/6, 0 ≤ k ≤ 5) while Eq.13 leads to 6 unique values for its imaginary part ((k·π - 1)/3, 1 ≤ k ≤ 6). Observing the first column of Fig.3, both the predicted number of curves and their tangential angles at the expansion point match the actual loci of exactness computed for $s_2$ and $s_3$. The predictions of Eqs.12,13 regarding $\log(1+z)$ and $\sin(z)/z$ are validated the same way and are depicted in the second and third rows of Fig.3.

# 6 Conclusions and perspectives

The present study has dealt with the general question of the accuracy of truncated Taylor series of analytic functions. In particular, the existence and properties of possible loci of exactness (i.e., points where the truncated series and the analytic function
<figure><img src="image_4.png" /><figcaption>Figure 3: Loci of exactness of $s_2(z)$ and $s_3(z)$ with their predicted tangent half-lines: (a) for the real part, (b) for the imaginary part.</figcaption></figure>
share the same values) have been investigated. Based on computational experiences, the existence of distinct loci of exactness for the real and imaginary parts of the truncated series has been observed. In particular, near the expansion point, the loci appear to aggregate on curves emanating from it in remarkable patterns. The number and tangential angles of these curves have been correlated to the exponent and the factor of the first term in the remainder associated with the truncated series. Beyond further mathematical investigations about these newly introduced curves of loci of exactness, practical applications can be made in the development of analytical and numerical methods to accurately evaluate series coefficients using data points along the curves.

## References

[1] B. Taylor. *Methodus Incrementorum Directa et Inversa*. Gul. Innys, London, 1715.

[2] R. Roy. *Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-first Century*. Cambridge University Press, New-York, 2011.

[3] R.E. Rodríguez, I. Kra, and J.P. Gilman. *Complex Analysis: In the Spirit of Lipman Bers*. Graduate Texts in Mathematics. Springer New York, New-York, 2013.

[4] J. Gray. *The Real and the Complex: A History of Analysis in the 19th Century*. Springer Undergraduate Mathematics Series. Springer International Publishing, Switzerland, 2015.

[5] NI Muskhelishvili. *Some Basic Problems of Mathematical Theory of Elasticity*. Noordhoff, Leyden, 1953.

[6] LMB da Costa Campos. *Complex Analysis with Applications to Flows and Fields*. Mathematics and Physics for Science and Technology. CRC Press, Boca Raton, 2010.

[7] FW Byron and RW Fuller. *Mathematics of Classical and Quantum Physics*. Dover Books on Physics. Dover Publications, New-York, 2012.

[8] K. Sato. *Complex Analysis for Practical Engineering*. Springer International Publishing, Switzerland, 2015.

[9] Gaëtan Hello. On the loci of exactness for truncated Williams crack-tip stress expansions. *International Journal of Fracture*, 248(1):81-94, Dec 2024.