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BSc: Mathematical Analysis II
Contents
- 1 MathematicalAnalysis II
- 1.1 Course Characteristics
- 1.2 Course Objectives Based on Bloom’s Taxonomy
- 1.3 Course Sections
- 1.3.1 Section 1
- 1.3.2 Topics covered in this section:
- 1.3.3 What forms of evaluation were used to test students’ performance in this section?
- 1.3.4 Typical questions for ongoing performance evaluation within this section
- 1.3.5 Typical questions for seminar classes (labs) within this section
- 1.3.6 Test questions for final assessment in this section
- 1.3.7 Section 2
- 1.3.8 Topics covered in this section:
- 1.3.9 What forms of evaluation were used to test students’ performance in this section?
- 1.3.10 Typical questions for ongoing performance evaluation within this section
- 1.3.11 Typical questions for seminar classes (labs) within this section
- 1.3.12 Test questions for final assessment in this section
- 1.3.13 Section 3
- 1.3.14 What forms of evaluation were used to test students’ performance in this section?
- 1.3.15 Typical questions for ongoing performance evaluation within this section
- 1.3.16 Test questions for final assessment in the course
- 1.3.17 Section 1
- 1.3.18 Topics covered in this section:
- 1.3.19 What forms of evaluation were used to test students’ performance in this section?
- 1.3.20 Typical questions for ongoing performance evaluation within this section
- 1.3.21 Typical questions for seminar classes (labs) within this section
- 1.3.22 Test questions for final assessment in this section
MathematicalAnalysis II
- Course name: Mathematical Analysis II
- Course number: BS-01
Course Characteristics
Key concepts of the class
- Multivariate calculus: derivatives, differentials, maxima and minima
- Multivariate integration
- Functional series. Fourier series
- Integrals with parameters
What is the purpose of this course?
The goal of the course is to study basic mathematical concepts that will be required in further studies. The course is based on Mathematical Analysis I, and the concepts studied there are widely used in this course. The course covers differentiation and integration of functions of several variables. Some more advanced concepts, as uniform convergence of series and integrals, are also considered, since they are important for understanding applicability of many theorems of mathematical analysis. In the end of the course some useful applications are covered, such as gamma-function, beta-function, and Fourier transform.
Course Objectives Based on Bloom’s Taxonomy
- What should a student remember at the end of the course?
By the end of the course, the students should be able to:
- find partial and directional derivatives of functions of several variables;
- find maxima and minima for a function of several variables
- use Fubini’s theorem for calculating multiple integrals
- calculate line and path integrals
- distinguish between point wise and uniform convergence of series and improper integrals
- decompose a function into Fourier series
- calculate Fourier transform of a function
- What should a student be able to understand at the end of the course?
By the end of the course, the students should be able to understand:
- how to find minima and maxima of a function subject to a constraint
- how to represent double integrals as iterated integrals and vice versa
- what the length of a curve and the area of a surface is
- properties of uniformly convergent series and improper integrals
- beta-function, gamma-function and their properties
- how to find Fourier transform of a function
- What should a student be able to apply at the end of the course?
By the end of the course, the students should be able to ...
- find multiple, path, surface integrals
- find the range of a function in a given domain
- decompose a function into Fourier series
Course evaluation
Course grade breakdown
| Proposed points | ||
|---|---|---|
| Test 1 | ||
| ? | ||
| 10 | ||
| Midterm | ||
| ? | ||
| 25 | ||
| Test 2 | ||
| ? | ||
| 10 | ||
| Participation | ||
| ? | ||
| 5 | ||
| Final exam | ||
| ? | ||
| 50 | ||
Grades range
Course grading range
| Proposed range | ||
|---|---|---|
| A. Excellent | ||
| 90-100 | ||
| 85-100 | ||
| B. Good | ||
| 75-89 | ||
| 65-84 | ||
| C. Satisfactory | ||
| 60-74 | ||
| 45-64 | ||
| D. Poor | ||
| 0-59 | ||
| 0-44 | ||
Resources and reference material
- Robert A. Adams, Christopher Essex (2017) Calculus. A Complete Course, Pearson
- Jerrold Marsden, Alan Weinstein (1985) Calculus (in three volumes; volumes 2 and 3), Springer
Course Sections
The main sections of the course and approximate hour distribution between them is as follows:
Course Sections
| Section | Section Title | Teaching Hours |
|---|---|---|
| 1 | ||
| Differential Analysis of Functions of Several Variables | ||
| 24 | ||
| 2 | ||
| Integration of Functions of Several Variables | ||
| 30 | ||
| 3 | ||
| Uniform Convergence of Functional Series. Fourier Series | ||
| 18 | ||
| 4 | ||
| Integrals with Parameter(s) | ||
| 18 | ||
| hline | ||
Section 1
Section title:
Differential Analysis of Functions of Several Variables
Topics covered in this section:
- Limits of functions of several variables
- Partial and directional derivatives of functions of several variables. Gradient
- Differentials of functions of several variables. Taylor formula
- Maxima and minima for functions of several variables
- Maxima and minima for functions of several variables subject to a constraint
What forms of evaluation were used to test students’ performance in this section?
| Yes/No | |
|---|---|
| Development of individual parts of software product code | |
| 0 | |
| Homework and group projects | |
| 1 | |
| Midterm evaluation | |
| 1 | |
| Testing (written or computer based) | |
| 1 | |
| Reports | |
| 0 | |
| Essays | |
| 0 | |
| Oral polls | |
| 0 | |
| Discussions | |
| 1 | |
Typical questions for ongoing performance evaluation within this section
- Find
lim
x → 0
lim
y → 0
u ( x ; y )
{\textstyle \lim \limits _{x\to 0}\lim \limits _{y\to 0}u(x;y)}
lim
y → 0
lim
x → 0
u ( x ; y )
{\textstyle \lim \limits _{y\to 0}\lim \limits _{x\to 0}u(x;y)}
lim
( x ; y ) → ( 0 ; 0 )
u ( x ; y )
{\textstyle \lim \limits _{(x;y)\to (0;0)}u(x;y)}
u ( x ; y )
x
2
y + x
y
2
x
2
− x y +
y
2
{\textstyle u(x;y)={\frac {x^{2}y+xy^{2}}{x^{2}-xy+y^{2}}}}
u ( x ; y )
ln
(
x +
x
2
y
2
)
{\textstyle u(x;y)=\ln \left(x+{\sqrt {x^{2}+y^{2}}}\right)}
u ( x ; y )
ln sin
x + 1
y
{\textstyle u(x;y)=\ln \sin {\frac {x+1}{\sqrt {y}}}}
u ( x ; y )
{\textstyle u(x;y)}
x
3
- 2
y
3
u
3
− 3 x y u + 2 y − 3
0
{\textstyle x^{3}+2y^{3}+u^{3}-3xyu+2y-3=0}
M ( 1 ; 1 ; 1 )
{\textstyle M(1;1;1)}
N ( 1 ; 1 ; − 2 )
{\textstyle N(1;1;-2)}
x
2
y
3
z
4
{\textstyle u=x^{2}y^{3}z^{4}}
,
2
x
+
3
y
+
4
z
=
18
{\textstyle 2x+3y+4z=18}
,
x
>
0
{\textstyle x>0}
,
y
>
0
{\textstyle y>0}
,
z
>
0
{\textstyle z>0}
;
2. u
=
x
−
y
+
2
z
{\textstyle u=x-y+2z}
,
x
2
+
y
2
+
2
z
2
=
16
{\textstyle x^{2}+y^{2}+2z^{2}=16}
;
3. u
=
∑
i
=
1
k
a
i
x
i
2
{\textstyle u=\sum \limits \_{i=1}^{k}a\_{i}x\_{i}^{2}}
,
∑
i
=
1
k
x
i
=
1
{\textstyle \sum \limits \_{i=1}^{k}x\_{i}=1}
,
a
i
>
0
{\textstyle a\_{i}>0}
;
Typical questions for seminar classes (labs) within this section
- Let us consider
u ( x ; y )
{
1 ,
x
y
2
,
0 ,
x ≠
y
2
.
{\textstyle u(x;y)={\begin{cases}1,&x=y^{2},\0,&x\neq y^{2}.\end{cases}}}
( x ; y ) → ( 0 ; 0 )
{\textstyle (x;y)\to (0;0)}
M ( 1 ; − 2 ; − 3 )
{\textstyle M(1;-2;-3)}
f
ln x y z
{\textstyle f=\ln xyz}
u ( x , y )
{\textstyle u(x,y)}
2
+
y
2
+
u
2
−
4
x
−
6
y
−
4
u
+
8
=
0
{\textstyle x^{2}+y^{2}+u^{2}-4x-6y-4u+8=0}
,
u
>
2
{\textstyle u>2}
;
2. x
3
−
y
2
+
u
2
−
3
x
+
4
y
+
u
−
8
=
0
{\textstyle x^{3}-y^{2}+u^{2}-3x+4y+u-8=0}
.
Find maxima and minima of functions subject to constraints: 1. u
x y 2 {\textstyle u=xy^{2}}
, x + 2 y − 1
0 {\textstyle x+2y-1=0}
; 2. u
x y + y z {\textstyle u=xy+yz}
, x 2 + y 2
2 {\textstyle x^{2}+y^{2}=2}
, y + z
2
{\textstyle y+z=2}
,
y
0
{\textstyle y>0}
.
Test questions for final assessment in this section
- Find all points where the differential of a function
f ( x ; y )
( 5 x + 7 y − 25 )
e
−
x
2
− x y −
y
2
{\textstyle f(x;y)=(5x+7y-25)e^{-x^{2}-xy-y^{2}}}
φ
f
(
x y
;
x
2
- y −
z
2
)
{\textstyle \varphi =f\left({\frac {x}{y}};x^{2}+y-z^{2}\right)}
2 x z
φ
x
- 2 y z
φ
y
(
2
x
2
- y
)
φ
z
= 0
{\textstyle 2xz\varphi _{x}+2yz\varphi _{y}+\left(2x^{2}+y\right)\varphi _{z}=0}
u
2
x
2
- 12 x y
y
2
{\textstyle u=2x^{2}+12xy+y^{2}}
x
2
- 4
y
2
= 25
{\textstyle x^{2}+4y^{2}=25}
. Find the maximum and minimum value of a function
4. u
(
y
2
−
x
2
)
e
1 −
x
2
y
2
{\textstyle u=\left(y^{2}-x^{2}\right)e^{1-x^{2}+y^{2}}}
x
2
y
2
≤ 4
{\textstyle x^{2}+y^{2}\leq 4}
Section 2
Section title:
Integration of Functions of Several Variables
Topics covered in this section:
- Z-test
- Double integrals. Fubini’s theorem and iterated integrals
- Substituting variables in double integrals. Polar coordinates
- Triple integrals. Use of Fubini’s theorem
- Spherical and cylindrical coordinates
- Path integrals
- Area of a surface
- Surface integrals
What forms of evaluation were used to test students’ performance in this section?
| Yes/No | |
|---|---|
| Development of individual parts of software product code | |
| 0 | |
| Homework and group projects | |
| 1 | |
| Midterm evaluation | |
| 1 | |
| Testing (written or computer based) | |
| 1 | |
| Reports | |
| 0 | |
| Essays | |
| 0 | |
| Oral polls | |
| 0 | |
| Discussions | |
| 1 | |
Typical questions for ongoing performance evaluation within this section
- Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration:
∬
D
f ( x ; y )
d x
d y
{\textstyle \iint \limits _{D}f(x;y),dx,dy}
D
{
( x ; y )
|
x
2
y
2
≤ 9 ,
x
2
- ( y
- 4
)
2
≥ 25
}
{\textstyle D=\left{(x;y)\left|x^{2}+y^{2}\leq 9,,x^{2}+(y+4)^{2}\geq 25\right.\right}}
I
∭
D
f ( x ; y ; z )
d x
d y
d z
{\textstyle I=\displaystyle \iiint \limits _{D}f(x;y;z),dx,dy,dz}
D
{\textstyle D}
x
0
{\textstyle x=0}
x
a
{\textstyle x=a}
y
0
{\textstyle y=0}
y
a x
{\textstyle y={\sqrt {ax}}}
z
0
{\textstyle z=0}
z
x + y
{\textstyle z=x+y}
∫
Γ
( x + y )
d s
{\textstyle \displaystyle \int \limits _{\Gamma }(x+y),ds}
Γ
{\textstyle \Gamma }
( 0 ; 0 )
{\textstyle (0;0)}
( 1 ; 0 )
{\textstyle (1;0)}
( 0 ; 1 )
{\textstyle (0;1)}
Typical questions for seminar classes (labs) within this section
- Change order of integration in the iterated integral
∫
0
2
d y
∫
y
4 −
y
2
f ( x ; y )
d x
{\textstyle \int \limits _{0}^{\sqrt {2}}dy\int \limits _{y}^{\sqrt {4-y^{2}}}f(x;y),dx}
0 ≤ z ≤
x
2
{\textstyle 0\leq z\leq x^{2}}
x + y ≤ 5
{\textstyle x+y\leq 5}
x − 2 y ≥ 2
{\textstyle x-2y\geq 2}
y ≥ 0
{\textstyle y\geq 0}
∬
G
f
(
x
2
y
2
)
d x
d y
{\textstyle \displaystyle \iint \limits _{G}f\left({\sqrt {x^{2}+y^{2}}}\right),dx,dy}
G
{
( x ; y )
|
x
2
y
2
≤ x ;
x
2
y
2
≤ y
}
{\textstyle G=\left{(x;y)\left|x^{2}+y^{2}\leq x;,x^{2}+y^{2}\leq y\right.\right}}
A
{\textstyle A}
B
{\textstyle B}
∫
Γ
(
x
4
- 4 x
y
3
)
d x +
(
6
x
2
y
2
− 5
y
4
)
d y
{\textstyle \displaystyle \int \limits _{\Gamma }\left(x^{4}+4xy^{3}\right),dx+\left(6x^{2}y^{2}-5y^{4}\right),dy}
A ( − 2 ; − 1 )
{\textstyle A(-2;-1)}
B ( 0 ; 3 )
{\textstyle B(0;3)}
Test questions for final assessment in this section
- Domain
G
{\textstyle G}
y
2 x
{\textstyle y=2x}
y
x
{\textstyle y=x}
y
2
{\textstyle y=2}
∬
G
f ( x )
d x
d y
{\textstyle \iint \limits _{G}f(x),dx,dy}
∬
G
f ( x ; y )
d x
d y
{\textstyle \displaystyle \iint \limits _{G}f(x;y),dx,dy}
G
{
( x ; y )
|
a
2
≤
x
2
y
2
≤ 4
a
2
;
|
x
|
− y ≥ 0
}
{\textstyle G=\left{(x;y)\left|a^{2}\leq x^{2}+y^{2}\leq 4a^{2};,|x|-y\geq 0\right.\right}}
∭
G
(
x
2
−
y
2
)
(
z +
x
2
−
y
2
)
d x
d y
d z
{\textstyle \displaystyle \iiint \limits _{G}\left(x^{2}-y^{2}\right)\left(z+x^{2}-y^{2}\right),dx,dy,dz}
G
{
( x ; y ; z )
|
x − 1 < y < x ;
1 − x < y < 2 − x ;
1 −
x
2
y
2
< z <
y
2
−
x
2
- 2 x
}
{\textstyle G=\left{(x;y;z)\left|x-1<y<x;,1-x<y<2-x;,1-x^{2}+y^{2}<z<y^{2}-x^{2}+2x\right.\right}}
∬
S
( 1 + 2 x )
d y
d z + ( 2 x + 3 y )
d z
d x + ( 3 y + 4 z )
d x
d y
{\textstyle \displaystyle \iint \limits _{S}(1+2x),dy,dz+(2x+3y),dz,dx+(3y+4z),dx,dy}
S
{\textstyle S}
x a
y b
z c
≤ 1
{\textstyle {\frac {x}{a}}+{\frac {y}{b}}+{\frac {z}{c}}\leq 1}
x ≥ 0
{\textstyle x\geq 0}
y ≥ 0
{\textstyle y\geq 0}
z ≥ 0
{\textstyle z\geq 0}
Section 3
Section title:
Uniform Convergence of Functional Series. Fourier Series
Topics covered in this section:
- Uniform and point wise convergence of functional series
- Properties of uniformly convergent series
- Fourier series. Sufficient conditions of convergence and uniform convergence
- Bessel’s inequality and Parseval’s identity.
What forms of evaluation were used to test students’ performance in this section?
| Yes/No | |
|---|---|
| Development of individual parts of software product code | |
| 0 | |
| Homework and group projects | |
| 1 | |
| Midterm evaluation | |
| 1 | |
| Testing (written or computer based) | |
| 1 | |
| Reports | |
| 0 | |
| Essays | |
| 0 | |
| Oral polls | |
| 0 | |
| Discussions | |
| 1 | |
Typical questions for ongoing performance evaluation within this section
- Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer.
∑
n
1
∞
e
− n
(
x
2
- 2 sin x
)
{\textstyle \sum \limits _{n=1}^{\infty }e^{-n\left(x^{2}+2\sin x\right)}}
Δ
1
= ( 0 ; 1 ]
{\textstyle \Delta _{1}=(0;1]}
![{\textstyle \Delta _{1}=(0;1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b80239da7d74486d76788367da82b612e233dcf6),
Δ
2
= [ 1 ; + ∞ )
{\textstyle \Delta _{2}=[1;+\infty )}
Δ
1
= ( 0 ; 1 )
{\textstyle \Delta _{1}=(0;1)}
Δ
2
= ( 1 ; + ∞ )
{\textstyle \Delta _{2}=(1;+\infty )}
f
n
( x )
n x
(
1 − x
)
n
{\textstyle f_{n}(x)=nx\left(1-x\right)^{n}}
[ 0 ; 1 ]
{\textstyle [0;1]}
![{\textstyle [0;1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f62f697cdfa218d158d17c6c527ce72134efed64)
f ( x )
{\textstyle f(x)}
lim
n → + ∞
∫
0
1
f
n
( x )
d x
lim
n → + ∞
∫
0
1
f ( x )
d x
{\textstyle \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f_{n}(x),dx=\lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f(x),dx}
[ − π ; π ]
{\textstyle [-\pi ;\pi ]}
![{\textstyle [-\pi ;\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8de05c42024ae085180a3cee60b8f13097d64f77)
{
1 ; cos k x ; sin k x
}
|
k
1
∞
{\textstyle \left.\left{1;\cos kx;\sin kx\right}\right|_{k=1}^{\infty }}
f ( x )
{
1 ,
0 ≤ x ≤ π ,
0 ,
− π ≤ x < 0.
{\textstyle f(x)={\begin{cases}1,;0\leq x\leq \pi ,\0,;-\pi \leq x<0.\end{cases}}}
f ( x )
{\textstyle f(x)}
[ − π ; π ]
{\textstyle [-\pi ;\pi ]}
{\textstyle f(x+\pi )=f(x)}
 then
a
2
k
−
1
=
b
2
k
−
1
=
0
{\textstyle a\_{2k-1}=b\_{2k-1}=0}
,
k
∈
N
{\textstyle k\in \mathbb {N} }
;
2. f
(
x
+
π
)
=
−
f
(
x
)
{\textstyle f(x+\pi )=-f(x)}
 then
a
0
=
0
{\textstyle a\_{0}=0}
,
a
2
k
=
b
2
k
=
0
{\textstyle a\_{2k}=b\_{2k}=0}
,
k
∈
N
{\textstyle k\in \mathbb {N} }
.
Typical questions for seminar classes (labs) within this section
- Show that sequence
f
n
( x )
n x
(
1 −
x
2
)
n
{\textstyle f_{n}(x)=nx\left(1-x^{2}\right)^{n}}
[ 0 ; 1 ]
{\textstyle [0;1]}
f ( x )
{\textstyle f(x)}
lim
n → + ∞
∫
0
1
f
n
( x )
d x ≠
lim
n → + ∞
∫
0
1
f ( x )
d x
{\textstyle \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f_{n}(x),dx\neq \lim \limits _{n\rightarrow +\infty }\int \limits _{0}^{1}f(x),dx}
f
n
( x )
x
3
1 n
sin
(
n x +
n π
2
)
{\textstyle f_{n}(x)=x^{3}+{\frac {1}{n}}\sin \left(nx+{\frac {n\pi }{2}}\right)}
R
{\textstyle \mathbb {R} }
(
lim
n → + ∞
f
n
( x )
)
′
≠
lim
n → + ∞
f
n
′
( x )
{\textstyle \left(\lim \limits _{n\rightarrow +\infty }f_{n}(x)\right)'\neq \lim \limits _{n\rightarrow +\infty }f'_{n}(x)}
cos α x
{\textstyle \cos \alpha x}
α ∉
Z
{\textstyle \alpha \notin \mathbb {Z} }
[ − π ; π ]
{\textstyle [-\pi ;\pi ]}
cot y
1 y
∑
k
1
∞
2 y
y
2
−
π
2
k
2
{\textstyle \cot y={\frac {1}{y}}+\sum \limits _{k=1}^{\infty }{\frac {2y}{y^{2}-\pi ^{2}k^{2}}}}
f ( x )
{\textstyle f(x)}
[ 0 ; π ]
{\textstyle [0;\pi ]}
![{\textstyle [0;\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eb0e57f19cbd824b48391708d429bf63d2e2a615)
f ( π − x )
f ( x )
{\textstyle f(\pi -x)=f(x)}
b
2
k
=
0
{\textstyle b\_{2k}=0}
,
k
∈
N
{\textstyle k\in \mathbb {N} }
;
2. if it is decomposed into Fourier series of cosines then
a
2
k
−
1
=
0
{\textstyle a\_{2k-1}=0}
,
k
∈
N
{\textstyle k\in \mathbb {N} }
.
f ( x )
{
1 ,
|
x
|
< α ,
0 ,
α ⩽
|
x
|
< π
{\textstyle f(x)={\begin{cases}1,;|x|<\alpha ,\0,;\alpha \leqslant |x|<\pi \end{cases}}}
σ
1
=
∑
k
=
1
∞
sin
2
k
α
k
2
{\textstyle \sigma \_{1}=\sum \limits \_{k=1}^{\infty }{\frac {\sin ^{2}k\alpha }{k^{2}}}}
 and
σ
2
=
∑
k
=
1
∞
cos
2
k
α
k
2
{\textstyle \sigma \_{2}=\sum \limits \_{k=1}^{\infty }{\frac {\cos ^{2}k\alpha }{k^{2}}}}
.
Test questions for final assessment in the course
- Find out whether the following functional series converge uniformly on the indicated intervals. Justify your answer.
∑
n
1
∞
x n +
n
n + x
ln
(
1 +
x
n
n
)
{\textstyle \sum \limits _{n=1}^{\infty }{\frac {xn+{\sqrt {n}}}{n+x}}\ln \left(1+{\frac {x}{n{\sqrt {n}}}}\right)}
Δ
1
= ( 0 ; 1 )
{\textstyle \Delta _{1}=(0;1)}
Δ
2
= ( 1 ; + ∞ )
{\textstyle \Delta _{2}=(1;+\infty )}
f
n
( x )
sin n x
n
{\textstyle f_{n}(x)={\frac {\sin nx}{\sqrt {n}}}}
R
{\textstyle \mathbb {R} }
f ( x )
{\textstyle f(x)}
lim
n → + ∞
f
n
′
( 0 ) ≠
f ′
( 0 )
{\textstyle \lim \limits _{n\rightarrow +\infty }f'_{n}(0)\neq f'(0)}
Section 1
Section title:
Integrals with Parameter(s)
Topics covered in this section:
- Definite integrals with parameters
- Improper integrals with parameters. Uniform convergence
- Properties of uniformly convergent integrals
- Beta-function and gamma-function
- Fourier transform
What forms of evaluation were used to test students’ performance in this section?
| Yes/No | |
|---|---|
| Development of individual parts of software product code | |
| 0 | |
| Homework and group projects | |
| 1 | |
| Midterm evaluation | |
| 1 | |
| Testing (written or computer based) | |
| 1 | |
| Reports | |
| 0 | |
| Essays | |
| 0 | |
| Oral polls | |
| 0 | |
| Discussions | |
| 1 | |
Typical questions for ongoing performance evaluation within this section
- Find out if
∫
0
1
(
lim
α → 0
2 x
α
2
(
α
2
x
2
)
2
)
d x
lim
α → 0
∫
0
1
2 x
α
2
(
α
2
x
2
)
2
d x
{\textstyle \displaystyle \int \limits _{0}^{1}\left(\lim \limits _{\alpha \to 0}{\frac {2x\alpha ^{2}}{\left(\alpha ^{2}+x^{2}\right)^{2}}}\right),dx=\lim \limits _{\alpha \to 0}\int \limits _{0}^{1}{\frac {2x\alpha ^{2}}{\left(\alpha ^{2}+x^{2}\right)^{2}}},dx}
φ
{\textstyle \varphi }
I ( α )
∫
0
π
/
2
ln
(
α
2
−
sin
2
φ
)
d φ
{\textstyle I(\alpha )=\int \limits _{0}^{\pi /2}\ln \left(\alpha ^{2}-\sin ^{2}\varphi \right),d\varphi }
α
1
{\textstyle \alpha >1}
∫
0
- ∞
e
− α x
cos 2 x
d x
{\textstyle \displaystyle \int \limits _{0}^{+\infty }e^{-\alpha x}\cos 2x,dx}
Δ
[ 1 ; + ∞ )
{\textstyle \Delta =[1;+\infty )}
π 2
{\textstyle {\frac {\pi }{2}}}
0
+
∞
sin
α
x
x
d
x
{\textstyle \int \limits \_{0}^{+\infty }{\frac {\sin }{\alpha x}}x\,dx}
,
α
≠
0
{\textstyle \alpha \neq 0}
;
2. ∫
0
+
∞
sin
x
−
x
cos
x
x
3
d
x
{\textstyle \int \limits \_{0}^{+\infty }{\frac {\sin x-x\cos x}{x^{3}}}\,dx}
.
Typical questions for seminar classes (labs) within this section
- Find out if
∫
0
1
(
∫
0
1
f ( x , α )
d α
)
d x
∫
0
1
(
∫
0
1
f ( x , α )
d x
)
d α
{\textstyle \displaystyle \int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha ),d\alpha \right),dx=\int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha ),dx\right),d\alpha }
f ( x ; α )
α − x
( α + x
)
3
{\textstyle f(x;\alpha )={\frac {\alpha -x}{(\alpha +x)^{3}}}}
Φ ′
( α )
{\textstyle \Phi '(\alpha )}
Φ ( α )
∫
1
2
e
α
x
2
x
d x
{\textstyle \Phi (\alpha )=\int \limits _{1}^{2}{\frac {e^{\alpha x^{2}}}{x}},dx}
α
{\textstyle \alpha }
I ( α )
∫
0
π
1
cos x
ln
1 + α cos x
1 − α cos x
d x
{\textstyle I(\alpha )=\int \limits _{0}^{\pi }{\frac {1}{\cos x}}\ln {\frac {1+\alpha \cos x}{1-\alpha \cos x}},dx}
|
α
|
< 1
{\textstyle |\alpha |<1}
{
1
,
|
x
|
≤
1
,
0
,
|
x
|
>
1
;
{\textstyle f(x)={\begin{cases}1,&|x|\leq 1,\\0,&|x|>1;\end{cases}}}
- Let
f ^
( y )
{\textstyle {\widehat {f}}(y)}
f ( x )
{\textstyle f(x)}
e
i α x
f ( x )
{\textstyle e^{i\alpha x}f(x)}
f ^
( y − α )
{\textstyle {\widehat {f}}(y-\alpha )}
α ∈
R
{\textstyle \alpha \in \mathbb {R} }
Test questions for final assessment in this section
- Find out if
∫
0
1
(
∫
0
1
f ( x , α )
d α
)
d x
∫
0
1
(
∫
0
1
f ( x , α )
d x
)
d α
{\textstyle \displaystyle \int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha ),d\alpha \right),dx=\int \limits _{0}^{1}\left(\int \limits _{0}^{1}f(x,\alpha ),dx\right),d\alpha }
f ( x ; α )
α
2
−
x
2
(
α
2
x
2
)
2
{\textstyle f(x;\alpha )={\frac {\alpha ^{2}-x^{2}}{\left(\alpha ^{2}+x^{2}\right)^{2}}}}
Φ ′
( α )
{\textstyle \Phi '(\alpha )}
Φ ( α )
∫
0
α
ln ( 1 + α x )
x
d x
{\textstyle \Phi (\alpha )=\int \limits _{0}^{\alpha }{\frac {\ln(1+\alpha x)}{x}},dx}
∫
− ∞
- ∞
cos α x
4 +
x
2
d x
{\textstyle \displaystyle \int \limits _{-\infty }^{+\infty }{\frac {\cos \alpha x}{4+x^{2}}},dx}
Δ
R
{\textstyle \Delta =\mathbb {R} }
f ( x )
{
1 ,
|
x
|
≤ τ ,
0 ,
|
x
|
τ ;
{\textstyle f(x)={\begin{cases}1,&|x|\leq \tau ,\0,&|x|>\tau ;\end{cases}}}
