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The big O notation is a mathematical notation to express various bounds concerning asymptotic behaviour of functions. It is often used in particular applications in physics, computer science, engineering and other applied sciences. For example, a typical context use in computer science is to express the complexity of algorithms.

More formally, if f and g are real valued functions of the real variable  t {\displaystyle t} then the notation  f ( t ) = O ( g ( t ) ) {\displaystyle f(t)=O(g(t))} indicates that there exist a real number T and a constant C such that  | f ( t ) | ≤ C | g ( t ) | {\displaystyle |f(t)|\leq C|g(t)|} for all  t > T . {\displaystyle t>T.}

Similarly, if  a n {\displaystyle a_{n}} and  b n {\displaystyle b_{n}} are two numerical sequences then  a n = O ( b n ) {\displaystyle a_{n}=O(b_{n})} means that  | a n | ≤ C | b n | {\displaystyle |a_{n}|\leq C|b_{n}|} for all n big enough.

The big O notation is also often used to indicate that the absolute value of a real valued function around some neighbourhood of a point is upper bounded by a constant multiple of the absolute value of another function, in that neigbourhood. For example, for a real number  t 0 {\displaystyle t_{0}} the notation  f ( t ) = O ( g ( t − t 0 ) ) {\displaystyle f(t)=O(g(t-t_{0}))} , where g(t) is a function which is continuous at t = 0 with g(0) = 0, denotes that there exists a real positive constant C such that  | f ( t ) | ≤ C | g ( t − t 0 ) | {\displaystyle |f(t)|\leq C|g(t-t_{0})|} on some neighbourhood N of  t 0 {\displaystyle t_{0}} .