Book: Introductio in analysin infinitorum

Overview
"Introductio in analysin infinitorum" (1748) by Leonhard Euler is a foundational textbook that reshaped how mathematicians think about functions and infinite processes. Its aim is to present a clear, coherent development of analysis as a discipline, moving from algebraic manipulation and series to a systematic study of transcendental functions. The book is notable for combining computational mastery with a unifying conceptual view that treats functions as central objects of study.

Structure and content
The work opens with a broad treatment of functions and their algebraic properties, then proceeds through the theory of infinite series, continued fractions, and the extraction of roots and logarithms by series methods. Trigonometric functions receive a detailed and elegant exposition: Euler develops their series expansions, relationships, and inverses, and casts them in the same analytic framework as exponential and logarithmic functions. Examples and worked computations are frequent, illustrating general principles by concrete manipulation of series, products, and transformations.

Key ideas and techniques
A central idea is the systematic use of infinite series and infinite products to define and study functions. Euler demonstrates how many functions of interest can be expanded, summed, and multiplied term by term, yielding formulas that link algebraic and transcendental quantities. He uses factorization of polynomials to motivate infinite product expressions for transcendental functions, and he exploits symmetries and substitutions to obtain concise functional identities. The treatment of the exponential, logarithmic, and trigonometric families emphasizes their interrelationships and presents analytic formulas that later became canonical.

Notation and conceptual innovations
Euler popularizes notation and viewpoints that make analysis more operational and accessible. He treats functions as entities that can be manipulated symbolically, emphasizes the parameter dependence of expressions, and advances the use of series as defining objects rather than mere approximations. The book contains celebrated formulas that connect seemingly disparate areas, exponential and trigonometric functions, infinite products for sine and cosine, and expansions that foreshadow later developments in complex analysis. The practical notation and problem-driven style helped disseminate these ideas across Europe.

Method and style
The tone is computational, guided by examples, clever substitutions, and inductive reasoning. Proofs are often heuristic by modern standards, relying on formal manipulations of infinite processes rather than on the epsilon–delta limits introduced in the following century. This approach makes the text highly effective for calculation and for suggesting general principles, even where rigorous justification was left to later analysts. The pedagogy emphasizes how to generate series and products, how to transform expressions, and how to use these tools to solve equations and analyze functions.

Historical impact and legacy
The "Introductio" established analysis as a coherent field and influenced generations of mathematicians. Its notions of function and analytic expansion fed directly into later rigorous theories of convergence and complex function theory. Many identities and techniques that Euler popularized became standard tools in mathematics, physics, and engineering. The work is regarded as a turning point that moved mathematical practice toward a broader, more flexible conception of functions and infinite processes, laying groundwork for the formalism and applications that followed.