Indian mathematicians made groundbreaking advances in trigonometry and infinite series. They created accurate sine tables, developed the versine function, and pioneered power series expansions for π and trigonometric functions.

The Kerala School, founded by Madhava, took these ideas further. They derived infinite series for π and arctangent, predating similar European discoveries by centuries. These innovations were crucial for solving complex astronomical problems and calculating mathematical constants.

Indian Trigonometry

Development of Sine Tables and Functions

• Indian mathematicians pioneered the creation of sine tables for astronomical calculations
• Aryabhata constructed sine tables with 24 entries at intervals of 3.75 degrees
• Sine function defined as half-chord of a circle, differing from modern definition
• Bhaskara I improved accuracy of sine tables using interpolation methods
• Madhava of Sangamagrama further refined sine calculations to 12 decimal places

Aryabhata's Contributions to Trigonometry

• Aryabhata introduced the versine (versin) function in his work Aryabhatiya
• Developed approximation methods for calculating sine values
• Established relationship between sine and cosine functions
• Utilized trigonometric functions for solving astronomical problems (planetary motions)
• Aryabhata's work laid foundation for later advancements in Indian trigonometry

Kerala School's Advancements

• Kerala School of Astronomy and Mathematics flourished from 14th to 16th centuries
• Madhava of Sangamagrama founded the Kerala School, making significant contributions
• Developed infinite series expansions for trigonometric functions
• Nilakantha Somayaji extended Madhava's work on planetary models
• Kerala School's achievements included accurate approximations of π and trigonometric functions

Infinite Series

Power Series Developments

• Indian mathematicians pioneered the use of power series expansions
• Madhava derived power series for π and arctangent function
• Jyesthadeva documented Madhava's discoveries in Yuktibhasa
• Power series allowed for more accurate calculations of mathematical constants
• Indian mathematicians utilized power series for solving complex astronomical problems

Madhava Series and Its Applications

• Madhava series refers to infinite series expansions discovered by Madhava of Sangamagrama
• Includes series for π, arctangent, sine, and cosine functions
• Madhava-Leibniz series for π: $\frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots$
• Madhava's series for arctangent: $\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots$
• These series provided more efficient methods for calculating trigonometric values

Precursors to Taylor Series

• Indian mathematicians developed techniques similar to modern Taylor series expansions
• Madhava's work on infinite series predated Taylor's discoveries by nearly 300 years
• Jyesthadeva documented methods for expanding functions as infinite series
• Kerala School mathematicians used series expansions to approximate trigonometric functions
• These developments laid groundwork for later European work on power series and calculus

Infinite Geometric Series

• Indian mathematicians explored properties of infinite geometric series
• Recognized the concept of convergence for certain infinite series
• Aryabhata discussed the sum of arithmetic and geometric progressions
• Brahmagupta provided formulas for summing arithmetic and geometric series
• Kerala School extended these concepts to infinite series, including alternating series