A» = dz :1م)د %3D e²dz ミ3 す
Riemann Sum
Riemann Sums is a special type of approximation of the area under a curve by dividing it into multiple simple shapes like rectangles or trapezoids and is used in integrals when finite sums are involved. Figuring out the area of a curve is complex hence this method makes it simple. Usually, we take the help of different integration methods for this purpose. This is one of the major parts of integral calculus.
Riemann Integral
Bernhard Riemann's integral was the first systematic description of the integral of a function on an interval in the branch of mathematics known as real analysis.
The question is about the
[ c ]is a circle with center 0 and radius 2
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no surprise that the value of an analytic function at a boundary (contour)
determines the function at all points inside the boundary.
19.1.4 Derivatives as Integrals
The CIF is a very powerful tool for working with analytic functions. One
of the applications of this formula is in evaluating the derivatives of such
functions. It is convenient to change the dummy integration variable to § and
write the CIF as
f(2) = f C) d£
2πί Jc ξ
1 [ f(E)
(19.9)
where C is a simple closed contour in the E-plane and z is a point within C.
By carrying the derivative inside the integral, we get
df
d [f(E) d£]
f(E) d£
2mi Jc (E – 2)²*
1
d
1
1
%3D
2ni de f LE) de
dz
dz
- Z
2ni
By repeated differentiation, we can generalize this formula to the nth deriva-
tive, and obtain
Theorem 19.1.11. The derivatives of all orders of an analytic function f(z)
exist in the domain of analyticity of the function and are themselves analytic
in that domain. The nth derivative of f(z) is given by
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