Mathematical Analysis Course Slides Notes
1. real number set
1.1. real number
Def: real number
Qua: real number basic quality
Qua: => real number absolute quality
Def:complex number
Qua: =>
Def: complex conjugate
Def: absolute value
Qua: operation
1.2. range & neibor & bound of set
Def: range & neibor of set
Def: exact bound of set
Qua: => exact bound
2. function
2.1. one variable function
Def: function
Note:
Def: bound & function
Def: mono function
- Theorm: mono => inverse mono
- Theorm: mono => inverse mono
Def: odd & function
Def: periodic & function
2.1.1. limit
2.1.1.1. limit
Def: limit of function to inf
Fig:
Def: limit of function to number
Fig:
Qua: 2 other functions larger & smaller => liim
Qua: ~ => liim
Qua: array => liim
Note:
Qua: f => liim
Example:
Example2
Qua: liim => only one
Qua: liim => bounded within a small range
Qua: liim => sign not change within a small range
Qua: liim => keep order within a small range
Qua: operation
Def: half way limit to number
Qua: mono => lim
Qua: mono => lim
2.1.1.2. O & o
Def: inf small
- Qua: operation
- Qua: operation
Def: o
Def: --
Def: O
Def: ~
2.1.2. continuous
2.1.2.1. continuous
Def: continuous at one point
- continuous = limit exist + (limit = value)
- def2
- def3
Qua: func => conti
Qua: func => conti
Qua: limit & continuous
Usage: (1) conti => limit
(2) limit do not require x0, conti does
(3) f and limit can exchange order
Qua: conti => keep range
Qua: conti => keep sign within range
Def: left/right continuous
Qua: suff & necc
Def: break point
Def: can be wiped break point (no def)
Def: jump break point (no lim)
Def: second break point (have def & lim but not equal)
2.1.2.2. continous function
Def: continuous function
Qua: conti => max/min
Qua: conti => f(x) = k
Qua: conti => inverse conti
Def: uniform continuous
Why: some conti func is too steep, like 1/x.
Note: Conti = choose a sigma(x, e) to make the gap < e
U-Conti = choose a sigma(e) to make the gap < e
Qua: suff
Def: several range continuous
2.1.3. derivation
2.1.3.1. def
Def: derivation
Qua: suff and necc
Qua: derble => some equation
Qua: derble => conti
Def: left/right der
Def: second order der
2.1.3.2. der function
Def: der function
Fig:
Qua: => der = k
Qua: => der = 0
Qua: => der = 0(k)
Qua: => der = g(x)
Qua: => der = c
Qua: mono => der >=0
Qua: => der = c
Qua: => liim der = c
Qua: operation
Def: max/ min
Qua: => min/max
Qua: => min/max
Qua: => min/max
2.1.3.3. differential
Def: differential
Qua: derible + (A=f') = diffble
Qua: 恒成立的等式
Qua: operation(ddx=0 from deltadeltax = 0)( 可微情况下 dx = deltax)
Def: high order diff
2.1.3.4. taylor expansion
Def:
Qua: => taylor
Qua: => taylor
Qua: => taylor
Qua: => taylor
2.1.4. Riemann integration
- Def:origin
Qua: conti => origin
Qua: operation
2.1.4.1. non fixed integration
Def: not fixed integration
Qua: => integration
Qua: => integration
Qua: devotion => integration
Qua: sin => integration
Qua: => integration
Qua: operation
2.1.4.2. fixed integration
Def:
Note:
Fig
Qua: => integration
Note:
Qua: => integration
Qua: => integration
Qua: => integration
Qua: => integration
2.1.4.2.1. integratable on [a,b]
Def: integratable on [a,b]
Qua: => intble
Qua: => intble
Qua: => intble
Qua: => intble
Qua: => intble
Qua: intble => bounded
Qua: intble =>
Qua: intble =>
Qua: intble =>
Qua: intble =>
Qua: operation
2.1.4.3. abnormal integration
Def: inf abnormal
Note:
Fig:
Qua: suff & necc of convergence
Qua: => convergecy
Qua: => convergency
Qua: operation
Def: break point abnormal integration
Note:
Qua: suff & necc
Qua: operation
2.2. multi variable function
2.2.1. limit
2.2.2. continuous
2.2.3. derivation
1 to 1 textbook
1 to X/M https://blog.csdn.net/weixin_38278334/article/details/83028794
X/M to X/M https://zhuanlan.zhihu.com/p/24863977 here is to mention that, the definition of 1 to X/M is different from X/M, we are more inclined to refer 1 to X/M as \(\triangle\), and the other as derivatives.
2.2.4. hidden variable
2.2.5. integration
2.2.5.1. parameter nomral integration
Def:
Qua: => conti
Qua: => diffble
Qua: => intble
Qua: => order changeble
2.2.5.2. parameter abnomral integration
2.2.5.2.1. def
Def:
Def: uniform convergency
Qua: suff & necc
Qua: => converge
Qua: converge => conti
Qua: converge => exchange order(conti)
Qua: converge => diffble
Qua: converge => intble
2.2.5.2.2. gamma & beta integration
Def: gamma
Fig:
Qua: gamma => diffble & conti
Qua: gamma => equation
Qua: gamma => tranform
Def: Beta
Qua: beta => conti
Qua: beta => equatiom
Qua: beta => transform
Qua: beta & gamma
2.2.5.3. curve integration
Def: first curve integration
Fig:
Qua: => integration
Qua: intble =>
Qua: intble => 路线无关性
Def: second curve integration
Note:
Qua: => integration
Qua: operation
2.2.5.4. muliple integrate
Def:
Qua: suff & neck
Def: multi integration
Note:
Qua: suff & necc
Qua: => intble
Qua: intble =>
Qua: => integration
Qua:Green, => integration
Qua: operation
3. array or series
3.1. number array/series
12
3.2. function array/series
3.2.1. function array
Def: function array
3.2.1.1. convergence
Def:
3.2.1.2. uni convergence
Def:
Note: visualization & why it's important
Qua:caughy =>
Note: same thing
Corollary:
3.2.1.3. f induced by function array
3.2.1.3.1. limit
Def: limit
Note; limit & limit exchange
3.2.1.3.2. continuous
Def: continuous
3.2.1.3.3. derivation
Def: derivable
Note: limit & deri exchange
3.2.1.3.4. Riemann integration
Def : integrable
Note: limit & intg exchange
3.2.2. function series
Def:
3.2.2.1. convergence
Def:
3.2.2.2. uni convergence
Def:
Qua: necc & suff
Note: same thing
Qua: some =>
Qua: some =>
Qua: some =>
3.2.2.3. f induced by function series
3.2.2.3.1. continuous
Def:
Note: sum & limit
3.2.2.3.2. derivation
Def:
Note: sum & deri
3.2.2.3.3. Riemann integration
Def:
Note: sum & intg
3.2.3. power series
Def:
3.2.3.1. convergence
Def:
Qua:abel =>
Note:
Qua: =>
Note: same thing
3.2.3.2. uni convergence
Def:
Qua: =>
Qua: =>
Qua:
3.2.3.3. f induced by power series
3.2.3.3.1. continuous
Def: continuous
3.2.3.3.2. deri & integral
Def:
Corollary:
3.2.3.4. computation of coefficient
Def: equavalence
Theorem: operation
3.2.3.5. taylor series
Def: taylor series
Usage: expansion without Rn
Def: generated by taylor series
Usage: when f = taylor series
Qua: same => exist
Qua: => uniqueness
3.2.4. fourier series
Def: tri series
Qua: convergence
Def: fourier series
Note:
Qua: =>
3.2.4.1. convergence
Def: smooth
Qua;
Def: convergence
Qua:
Qua: 2 \(\pi\) -> 2l
Qua: single / not
