We are excited to announce that the second edition of the first volume of Analysis with Mathematica was published by de Gruyter. You can order the book from: Filipuk, Galina and Kozłowski, Andrzej. “Frontmatter”. Analysis with Mathematica®: Single Variable Calculus, De Gruyter, 2026, pp. I-IV. ISBN 978-3-11-152214-2. https://doi.org/10.1515/9783111533063-201
​https://amzn.eu/d/07O0xYY8
The new edition of the book contains numerous additions. New material has been included in every chapter, mostly in the form of new examples or additional explanations of mathematical concepts as well as the Wolfram Programming Language. We used Mathematica’s version 14.2. Brief descriptions of many Wolfram Language functions, which we believe may be useful to the readers of this text have also been added. In some chapters we have also included more extensive discussions of certain aspects of the programming language that we found relevant. However, the most novel addition is the entirely new chapter, devoted to a new feature of Mathematica (and of Wolfram Language), which only appeared after the first edition was published. This new feature is the ability to use Artificial Intelligence, or, more accurately, Large Language Model chat bots from within Mathematica. This is an exciting new development, which will play a big part in the future of mathematics education and, although it is still in its infancy, it is growing up remarkably fast and should be of interest to both students and educators.
The main objective of this book is to convince the readers that Mathematica can be used effectively as an aid in solving mathematical problems or at least to inspire the idea of a solution or as a teaching aid. Our approach is to describe some of the theory and to guide the readers through solutions to a number of classroom problems. Most of these problems come from the course of analysis for computer science students, given at the University of Warsaw. The problems we chose were the ones in which Mathematica could be genuinely useful in a way that did not involve simply applying some standard algorithm or plotting a graph, but which required the students to think and come up with some mathematical idea. Sometimes this idea is a guess inspired by a Mathematica calculation or a visual representation of the problem. In some special cases Mathematica can only verify this guess and sometimes it can prove its correctness in full generality. In all such cases we normally still want to obtain a rigorous mathematical proof. In some cases we will
provide such proofs in detail, although sometimes we will only sketch them or limit ourselves to a special case, leaving the details and greater generality to the readers.
This book offers a thorough exploration of foundational topics in mathematical analysis (single-variable calculus), paired with extensive demonstrations of how to use Mathematica to handle these concepts computationally. It begins by covering number systems—ranging from sets, domains, and assumptions to real and complex numbers, infinities, and interval arithmetic—and illustrates principles like mathematical induction and telescoping sums. Subsequent chapters delve into recursive sequences, discrete dynamical systems such as the Fibonacci sequence, and methods like the secant method. The text then systematically investigates series, examining convergence tests, conditional versus absolute convergence, Cauchy products, and power series. Turning to limits of functions and continuity, the book discusses one-sided limits, continuous and discontinuous functions, and key theorems on continuous functions, followed by an in-depth treatment of differentiation—the definition, rules, and applications—and convexity. It continues with sequences and series of functions, detailing Taylor polynomials, convergence properties, and approximation techniques. Integration is handled next, exploring the Risch algorithm, the Riemann integral, improper integrals, and the use of Mathematica’s Integrate and NIntegrate functions. Finally, the book concludes with insights on leveraging large language models (LLMs) within Mathematica, showcasing examples and potential pitfalls when integrating modern AI techniques into mathematical workflows. Additionally, you can find out more about our exploration of LLMs withing Mathematica in a recent paper “Evolution of ChatGPT in Mathematica: a user experience” published in the European Mathematical Society Magazine https://ems.press/journals/mag/articles/14299459
We have tried to keep the book self-contained; however, some prior knowledge of mathematical analysis is required for a better understanding of the book. We hope that students and educators will benefit from this book a lot.
Below is a sample material from the book.

Subsection 2.3 Example

Find the limit of the sequence
2
,
2+
2
,
2+
2+
2+...
We can write this sequence using the function Nest:
In[]:=
Nest[Sqrt[2+#1]&,0,4]
Out[]=
2+
2+
2+
2
In[]:=
N[%]
Out[]=
1.99037
or using the recurrence and dynamic programming
In[]:=
a[1]:=0
In[]:=
a[n_]:=a[n]=Sqrt[a[n-1]+2]
In[]:=
a[4]
Out[]=
2+
2+
2
In[]:=
N[a[20]]
Out[]=
2.
The sequence is increasing and bounded. This can be proved by indunction since
a
1
=0<2
and assuming that
a
n
<=2
we have
a
n+1
=
a
n
+2<=
2+2
=2.
Hence the limit exists and is equal to
a
, which can be found from
In[]:=
a/.Solve[a==Sqrt[a+2],a][[1]]
Out[]=
2
since sequences
a
n
and
a
n-1
have the same limit.

From Section 4.8 Uniform continuity and the Lipschitz property

Theorem 13. Every continuous function fon a bounded closed interval [a,b] is uniformly continuous therein.
We can demonstrate this by restricting the function
x→
2
x
to the unit interval:
In[]:=
f[x_]:=
2
x
In[]:=
Reduce[ForAll[e,e>0,Exists[d,d>0,ForAll[{x,y},0<=x<=1&&0<=y<=1&&Abs[x-y]<d,Abs[f[x]-f[y]]<e]]]]
Out[]=
True
A function
f:A→ℝ
satisfies the Lipschitz condition if there exists a constant
c>0
such that
|f(x)−f(y)|≤c|x−y|
. Obviously, every function that satisfies the Lipschitz condition is uniformly continuous.
Let us show that the function
x→
x
does not satisfy the Lipschitz condition on [0, ∞) but does so on [1, ∞):
In[]:=
Reduce[Exists[c,c>0,ForAll[{x,y},x>=0&&y>=0,Abs[Sqrt[x]-Sqrt[y]]<=c*Abs[x-y]]],Reals]
Out[]=
False
In[]:=
Reduce[Exists[c,c>0,ForAll[{x,y},x>=1&&y>=1,Abs[Sqrt[x]-Sqrt[y]]<=c*Abs[x-y]]],Reals]
Out[]=
True
In fact we can actually, instead of deciding whether such a constant cexists, find its value by removing the function Exists from the expression above:
In[]:=
Reduce[ForAll[{x,y},x>=1&&y>=1,Abs[Sqrt[x]-Sqrt[y]]<=c*Abs[x-y]],Reals]
Out[]=
c≥
1
2
Note that this time we needed to add the domain Reals to Reduce:
In[]:=
Reduce[Exists[c,c>0,ForAll[{x,y},x>=0&&y>=0,Abs[Sqrt[x]-Sqrt[y]]<=c*Abs[x-y]]]]
Reduce
:Reduce was unable to prove that a radical of an expression containing only real variables and parameters is real-valued. If you are interested only in solutions for which all radicals contained in the input are real-valued, use Reduce with domain argument Reals.
Out[]=
Reduce
∃
c,c>0
∀
{x,y},x≥0&&y≥0
Abs
x
-
y
≤cAbs[x-y]
This happens whenever functions that can take non-real values, such as roots, appear in an expression to which Reduce is applied.
The reader should prove all the above results by hand. See also [14] for other examples.

CITE THIS NOTEBOOK

Analysis with Mathematica: Single Variable Calculus vol .1 second edition now available ​
by GalinaFilipuk & Andrzej Kozłowski
Wolfram Community, STAFF PICKS, April 2, 2026
​https://community.wolfram.com/groups/-/m/t/3673437