Transcendental functions are mathematical functions that cannot be expressed as a finite sequence of algebraic operations (addition, subtraction, multiplication, division, and taking roots). Common examples include the exponential function, trigonometric functions, and logarithmic functions. These functions often arise in various fields such as physics, engineering, and economics due to their ability to model complex phenomena. Unlike algebraic functions, which can be defined by polynomial equations, transcendental functions are typically defined through infinite series or integrals. Their unique properties make them essential in calculus and analysis, where they help solve differential equations and describe growth patterns, oscillations, and other dynamic systems.