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A function f is holomorphic in an open subset U of the complex plane if it is complex differentiable at every point in U. This means that the limit of the difference quotient, (f(z+h) - f(z))/h, exists as h approaches 0 for all z in U. This condition is significantly more restrictive than real differentiability and implies that the function is analytic.

The foundations of holomorphic functions were laid in the 19th century by mathematicians such as Cauchy, Riemann, Weierstrass, and Goursat. Cauchy's integral theorem and formula were pivotal, demonstrating the deep connection between a function's values and its behavior. Riemann's work on complex manifolds and conformal mappings further expanded the field, while Weierstrass's rigor solidified the concept of analyticity as equivalent to holomorphicity.

Holomorphic functions are fundamental due to their rich theoretical properties and wide-ranging applications. They are instrumental in solving differential equations, analyzing potential theory, and understanding conformal mappings, which preserve angles and are crucial in geometry and physics. Their applications extend to aerodynamics (potential flow), electromagnetism, quantum field theory, and signal processing.

The differentiability of a holomorphic function implies that it is infinitely differentiable and can be represented by a convergent power series in a neighborhood of each point (i.e., it is analytic). This is captured by the Cauchy-Riemann equations, which provide necessary and sufficient conditions for a function to be holomorphic. Properties like the maximum modulus principle and Liouville's theorem highlight their unique behavior.

Classic examples include polynomials, the exponential function (e^z), trigonometric functions (sin z, cos z), and the logarithm (log z, with appropriate branch cuts). These functions are used to model phenomena like wave propagation, heat distribution, and fluid flow. In engineering, they are employed in control theory, circuit analysis, and the design of antennas and waveguides.