I see. Thank you pwsnafu.pwsnafu said:Transcendental means in this context "to transcend algebra". At the time algebra meant studying polynomials. For example the fundamental theorem of algebra is a theorem about the roots of polynomials. Similarly an algebraic number is a number which is the root of a polynomial (usually with rational coefficients).
Transcendental functions are functions that cannot be expressed as polynomials, usually they are expressed as an infinite series. Even today, algebra is concerned with finite number of operations while analysis is concerned with infinite operations.
Transcendental functions are called so because they are functions that cannot be expressed as a finite combination of algebraic functions. These functions involve complex numbers and often involve infinite series. The term "transcendental" comes from the idea that these functions go beyond the scope of traditional algebraic functions.
Some examples of transcendental functions include exponential functions, logarithmic functions, trigonometric functions, and hyperbolic functions. These functions are commonly used in mathematics, physics, and engineering to model various natural phenomena.
The key difference between transcendental functions and algebraic functions is that algebraic functions can be expressed as a finite combination of rational functions and algebraic operations, such as addition, subtraction, multiplication, and division. Transcendental functions, on the other hand, cannot be expressed in this way and often require more complex methods, such as infinite series, to be evaluated.
Transcendental functions are important because they allow us to model and analyze a wide range of natural and physical phenomena, such as the growth of populations, the movement of waves, and the behavior of electrical circuits. They also have numerous applications in fields like engineering, physics, and economics.
The term "transcendental functions" was first used by Swiss mathematician Leonhard Euler in the 18th century. However, the concept of transcendental functions dates back to ancient times, with early civilizations using transcendental functions, such as trigonometric functions, to solve practical problems.