The Abstract Of Mathematical Analysis I
于2020年11月8日2020年11月8日由Sukuna发布
1. Limits
Two important limit
Definition 3. inferior limit and superior limit
Theorem 2. Stolz
Let
and
be two sequences of real numbers. Assume that
is a strictly monotone and divergent sequence (i.e. strictly increasing and approaching
, or strictly decreasing and approaching
) and the following limit exists:
Then, the limit
Theorem 3. Toeplitz limit theorem
Supports that
,
and
if
, let
, s.t.
By using
, we can quickly infer The Cauchy proposition theorem. By using
, we can quickly infer The Stolz theorem.
Stirling’s formula
Specifying the constant in the
error term gives
, yielding the more precise formula:
2. Continuity
Definition 0
A function
is continuous at the point
, if for any neighbourhood
of its value
at a there is a neighbourhood
of a whose image under the mapping
is contained in
.
3. Differential calculus
Definition 0
The number
is called the derivative of the function
at
.
Definition 1
A function
defined on a set
is differentiable at a point x ∈ E that is a limit point of E if
, where
is a linear function in
and
as
,
.
Definition 2
The function
of Definition 1, which is linear in
, is called the differential of the function
at the point
and is denoted
or
. Thus,
.
We obtain
We denote the set of all such vectors by
or
. Similarly, we denote by
or
the set of all displacement vectors from the point
along the y-axis. It can then be seen from the definition of the differential that the mapping
The derivative of an inverse function
If a function
is differentiable at a point x0 and its differential
is invertible at that point, then the differential of the function
inverse to
exists at the point
and is the mapping
inverse to
.
The derivative of some common function formula
L’Hôpital’s rule
The theorem states that for functions
and
which are differentiable on an open interval
except possibly at a point
contained in
, if
Taylor’s theorem
Let
be an integer and let the function
be
times differentiable at the point
. Then there exists a function
such that ,
and,
prove:
q.e.d
remainder term
using little
notation,
(The Peano remainder term)
The Lagrange form remainder term( Mean-value forms)
4. Integral
Antiderivative
Definition
In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function
is a differentiable function
whose derivative is equal to the original function
Suppose
, the notation is
So all the antiderivative of
become a family set
. also the equation below is obviously.
Theorem: Integration by parts
Example: Wallis product
the Wallis product for
, published in 1656 by John Wallis states that
Prove:
so that:
so that:
Simplify the Polynomial and Integral
If
and
is a proper fraction, there exists a unique representation of the fraction
in the form
and if
and
are polynomials with real coefficients and
there exists a unique representation of the proper fraction
in the form
where
and
are real numbers.
and with these formulas below:
And from that we get the recursion:
Primitives of the Form
We make the change of variable
. Since:
so that
It follows that
not only
can to do this, but here are a lot of formula:
,
,
,
Integration
Riemann Sums
partition
A partition P of a closed interval
,
, is a finite system of points
of the interval such that
.
If a function
is defined on the closed interval
and
is a partition with distinguished points on this closed interval, the sum
where
, is the Riemann sum of the function
corresponding to the partition
with distinguished points on
.
The largest of the lengths of the intervals of the partition
, denoted
, is called the mesh of the partition.
we define:
Integral mean value theorem
If
is a continuous function on the closed, bounded interval
, then there is at least one number
in
for which
The second Integral mean value theorem
If
are continuous functions on the closed, bounded interval
,
is monotonous on
, then there is at least one number
in
for which
Newton-Leibniz formula
Let
be a continuous real-valued function defined on a closed interval
. Let
be the function defined, s.t.
Substitution Rule For Definite Integrals
Suppose
and
, s.t.
