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Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration. It consists of two parts:
If f of x is continuous on the interval [a, b] and F of x is its antiderivative, then:
integral from a to b of f of x with respect to x equals F of b minus F of a.
If F of x is defined as an integral function:
F of x equals integral from a to x of f of t with respect to t,
then F of x is differentiable, and its derivative is the original function:
d by dx of F of x equals f of x.
Taylor Series Expansion
A function f of x can be expressed as an infinite Taylor series around x equals a:
summation from n equals zero to infinity of (nth derivative of f at a) divided by (n factorial) times (x minus a) to the power of n.
For example, the Taylor series expansion of e to the power of x at x equals zero is:
summation from n equals zero to infinity of (x to the power of n) divided by (n factorial), which expands as 1 plus x plus (x squared divided by 2 factorial) plus (x cubed divided by 3 factorial) and so on.
Complex Line Integrals
In complex analysis, contour integrals play a crucial role. The contour integral of a function f of z along a curve C is given by:
closed contour integral along C of f of z with respect to z.
A key result is Cauchy's Integral Formula:
f of a equals (1 divided by 2 pi i) times the closed contour integral along C of (f of z divided by (z minus a)) with respect to z,
which holds if f of z is analytic inside and on C, and a is within C.