Islamic science and mathematics experienced a boom during middle ages, contributions in mathematics, mechanics, astronomy and medicine were especially prominent, and had a deep impact on Renaissance Europe. There are many historical articles on Muslim Heritage: Science, see also links on Muslim Philosophers, Mathematicians & Scientists.
In mathematics aside from Khwarizmi's famous Hisab Al-Jabr wal Muqabala we have Omar Khayyam's Treatise on the Demonstration of Problems of Algebra, that has complete classification and geometric solution of all cubics. His also Explanations of the Difficulties in the Postulates of Euclid is considered the first work on the parallel postulate that avoids circular attempts to prove it, and anticipates a path to the hyperbolic geometry using a quadrilateral, later rediscovered by Saccheri and named after him in the West. Al-Karaji's s Al-Fakhri Fil-Jabr wal-Muqabala (Glorious on Algebra) introduced notations for all positive and negative powers of a variable and gave laws of exponents among them. Al-Samawal's Shining Book on Calculation gave the first version of long/synthetic division of polynomials. Al-Tusi's Treatise On The Quadrilateral was the most authoritative medieval work on trigonometry, and the first one to treat it independently of astronomy. See Kadyrov's Muslim Contributions to Mathematics for more.
Meragha school reformed epicyclic astronomy of Ptolemy making it consistent with (Aristotelian) dynamics, improving its precision and computational techniques. Al-Tusi’s Tadhkira i ilm al-Haya (Memoir on Astronomy) was a thorough critique of Ptolemy, especially of eccentrics and equants that appeared as ad hoc tricks. Al-Tusi found a way to model oscillatory linear motion with epicycles (the so-called Tusi couple), and used it in place of equants. In the Final Quest Concerning the Rectification of Principles al-Shatir eliminated eccenters and equants completely, without sacrificing precision, by mounting epicycles on epicycles. Although there is no direct confirmation, numerical coincidences strongly suggest that Copernicus used al-Shatir's models in his heliocentric reformulation.
In his influential commentary on Aristotle's physics Avempace revived the impetus theory of Philoponus (Avicenna discussed it as well), an alternative to Aristotle's theory of projectile motion, and an inspiration for momentum in modern dynamics. His commentary made its way to Europe, where the impetus theory was taken up by Buridan, Oresme and the merton school, who eventually influenced Galileo and Descartes.