The Abstract Of Mathematical Analysis I

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.