Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.
Analisis matematis adalah cabang ilmu matematika yang mencakup teori turunan, integral, ukuran, limit, deret, dan analisis fungsional.
Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis.
Analisis ini dikembangkan dari kalkulus, yang mencakup konsep dasar dan teknik analisis.
Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).
Namun, analisis ini dapat diterapkan di seluruh ruang objek matematika yang memiliki definisi kedekatan (ruang topologi) atau jarak tertentu di antara objek (ruang metrik).
Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids.
Menyusul matematikawan Yunani seperti Eudoxus and Archimedes menjadikannya lebih eksplisit, tetapi tidak formal, menggunakan konsep limit dan konvergensi saat mereka menggunakan metode penghabis untuk menghitung luas bangun datar dan volume bangun ruang.
In the 12th century, the Indian mathematician Bhāskara II used infinitesimal and used what is now known as Rolle's theorem.
Di India, matematikawan abad ke-12 Bhāskara II memberi contoh tentang turunan dan menggunakan konsep seperti yang sekarang dikenal dengan nama Teorema Rolle.
His followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century.
Pengikutnya di mazhab astronomi dan matematika Kerala melanjutkan karyanya hingga abad ke-16.
During this period, calculus techniques were applied to approximate discrete problems by continuous ones.
Dalam periode ini, teknik kalkulus digunakan untuk memperkirakan masalah diskret melalui pendekatan numerik.
In the 18th century, Euler introduced the notion of a mathematical function.
Pada abad ke-18, Euler memperkenalkan konsep fungsi matematika.
Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become widely known until the 1870s.
Analisis yang sesungguhnya mulai muncul sebagai subjek independen saat Bernard Bolzano memperkenalkan definisi kontinuitas pada tahun 1816, tetapi hasil kerjanya tidak dikenal luas sampai tahun 1870.
In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler.
Pada 1821, Cauchy mulai menempatkan kalkulus pada landasan yang kuat dengan menolak prinsip aljabar umum yang secara luas digunakan dalam karya sebelumnya, terutama oleh Euler.
Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals.
Sebaliknya, Cauchy merumuskan kalkulus dalam bentuk ide geometris dan infinitesimal.
He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis.
Ia juga memperkenalkan konsep urutan cauchy, dan memulai teori formal analisis kompleks.
Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis.
Poisson, Liouville, Fourier dan lainnya mempelajari persamaan diferensial parsial dan analisis harmonik.
The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis.
Kontribusi para matematikawan ini termasuk juga Weierstrass, mengembangkan pendekatan definisi limit (ε, δ) membuka babak baru bidang analisis matematis modern.
Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions.
Dedekind kemudian menyusun bilangan riil dengan potongan dedekind, di mana bilangan irasional didefinisikan secara formal, yang berfungsi untuk mengisi "celah" di antara bilangan rasional, sehingga menciptakan satu set kontinum bilangan riil yang telah dikembangkan oleh Simon Stevin.
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