Difference between revisions of "BSc:Mathematical Analysis Science"
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| − | = MathematicalAnalysis II = |
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| − | |||
| − | * <span>'''Course name:'''</span> Mathematical Analysis II |
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| − | * <span>'''Course number:'''</span> BS-01 |
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| − | |||
| − | == Course Characteristics == |
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| − | |||
| − | === Key concepts of the class === |
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| − | |||
| − | * Multivariate calculus: derivatives, differentials, maxima and minima |
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| − | * Multivariate integration |
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| − | * Functional series. Fourier series |
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| − | * Integrals with parameters |
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| − | |||
| − | === What is the purpose of this course? === |
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| − | |||
| − | 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. |
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| − | |||
| − | === Course objectives based on Bloom’s taxonomy === |
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| − | |||
| − | === - What should a student remember at the end of the course? === |
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| − | |||
| − | By the end of the course, the students should be able to: |
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| − | |||
| − | * find partial and directional derivatives of functions of several variables; |
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| − | * find maxima and minima for a function of several variables |
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| − | * use Fubini’s theorem for calculating multiple integrals |
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| − | * calculate line and path integrals |
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| − | * distinguish between point wise and uniform convergence of series and improper integrals |
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| − | * decompose a function into Fourier series |
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| − | * calculate Fourier transform of a function |
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| − | |||
| − | === - What should a student be able to understand at the end of the course? === |
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| − | |||
| − | By the end of the course, the students should be able to understand: |
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| − | |||
| − | * how to find minima and maxima of a function subject to a constraint |
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| − | * how to represent double integrals as iterated integrals and vice versa |
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| − | * what the length of a curve and the area of a surface is |
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| − | * properties of uniformly convergent series and improper integrals |
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| − | * beta-function, gamma-function and their properties |
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| − | * how to find Fourier transform of a function |
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| − | |||
| − | === - What should a student be able to apply at the end of the course? === |
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| − | |||
| − | By the end of the course, the students should be able to ... |
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| − | |||
| − | * find multiple, path, surface integrals |
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| − | * find the range of a function in a given domain |
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| − | * decompose a function into Fourier series |
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| − | |||
| − | === Course evaluation === |
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| − | |||
| − | {| |
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| − | |+ Course grade breakdown |
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| − | ! |
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| − | ! |
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| − | !align="center"| '''Proposed points''' |
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| − | |- |
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| − | | Test 1 |
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| − | | ? |
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| − | |align="center"| 10 |
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| − | |- |
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| − | | Midterm |
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| − | | ? |
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| − | |align="center"| 25 |
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| − | |- |
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| − | | Test 2 |
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| − | | ? |
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| − | |align="center"| 10 |
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| − | |- |
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| − | | Participation |
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| − | | ? |
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| − | |align="center"| 5 |
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| − | |- |
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| − | | Final exam |
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| − | | ? |
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| − | |align="center"| 50 |
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| − | |} |
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| − | |||
| − | === Grades range === |
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| − | |||
| − | {| |
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| − | |+ Course grading range |
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| − | ! |
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| − | ! |
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| − | !align="center"| '''Proposed range''' |
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| − | |- |
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| − | | A. Excellent |
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| − | | 90-100 |
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| − | |align="center"| 85-100 |
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| − | |- |
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| − | | B. Good |
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| − | | 75-89 |
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| − | |align="center"| 65-84 |
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| − | |- |
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| − | | C. Satisfactory |
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| − | | 60-74 |
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| − | |align="center"| 45-64 |
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| − | |- |
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| − | | D. Poor |
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| − | | 0-59 |
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| − | |align="center"| 0-44 |
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| − | |} |
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| − | |||
| − | === Resources and reference material === |
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| − | |||
| − | * Robert A. Adams, Christopher Essex (2017) Calculus. A Complete Course, Pearson |
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| − | * Jerrold Marsden, Alan Weinstein (1985) Calculus (in three volumes; volumes 2 and 3), Springer |
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| − | |||
| − | == Course Sections == |
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| − | |||
| − | The main sections of the course and approximate hour distribution between them is as follows: |
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| − | |||
| − | {| |
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| − | |+ Course Sections |
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| − | !align="center"| '''Section''' |
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| − | ! '''Section Title''' |
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| − | !align="center"| '''Teaching Hours''' |
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| − | |- |
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| − | |align="center"| 1 |
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| − | | Differential Analysis of Functions of Several Variables |
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| − | |align="center"| 24 |
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| − | |- |
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| − | |align="center"| 2 |
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| − | | Integration of Functions of Several Variables |
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| − | |align="center"| 30 |
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| − | |- |
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| − | |align="center"| 3 |
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| − | | Uniform Convergence of Functional Series. Fourier Series |
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| − | |align="center"| 18 |
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| − | |- |
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| − | |align="center"| 4 |
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| − | | Integrals with Parameter(s) |
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| − | |align="center"| 18 |
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| − | |- |
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| − | |align="center"| hline |
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| − | | |
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| − | |align="center"| |
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| − | |} |
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| − | |||
| − | === Section 1 === |
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| − | |||
| − | ==== Section title: ==== |
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| − | |||
| − | Differential Analysis of Functions of Several Variables |
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| − | |||
| − | === Topics covered in this section: === |
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| − | |||
| − | * Limits of functions of several variables |
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| − | * Partial and directional derivatives of functions of several variables. Gradient |
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| − | * Differentials of functions of several variables. Taylor formula |
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| − | * Maxima and minima for functions of several variables |
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| − | * Maxima and minima for functions of several variables subject to a constraint |
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| − | |||
| − | === What forms of evaluation were used to test students’ performance in this section? === |
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| − | |||
| − | <div class="tabular"> |
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| − | |||
| − | <span>|a|c|</span> & '''Yes/No'''<br /> |
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| − | Development of individual parts of software product code & 0<br /> |
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| − | Homework and group projects & 1<br /> |
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| − | Midterm evaluation & 1<br /> |
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| − | Testing (written or computer based) & 1<br /> |
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| − | Reports & 0<br /> |
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| − | Essays & 0<br /> |
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| − | Oral polls & 0<br /> |
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| − | Discussions & 0<br /> |
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| − | |||
| − | |||
| − | |||
| − | </div> |
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| − | === Typical questions for ongoing performance evaluation within this section === |
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| − | |||
| − | # Find <math display="inline">\lim\limits_{x\to0}\lim\limits_{y\to0}u(x;y)</math>, <math display="inline">\lim\limits_{y\to0}\lim\limits_{x\to0}u(x;y)</math> and <math display="inline">\lim\limits_{(x;y)\to(0;0)}u(x;y)</math> if <math display="inline">u(x;y)=\frac{x^2y+xy^2}{x^2-xy+y^2}</math>. |
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| − | # Find the differential of a function: (a) <math display="inline">u(x;y)=\ln\left(x+\sqrt{x^2+y^2}\right)</math>; (b) <math display="inline">u(x;y)=\ln\sin\frac{x+1}{\sqrt y}</math>. |
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| − | # Find the differential of <math display="inline">u(x;y)</math> given implicitly by an equation <math display="inline">x^3+2y^3+u^3-3xyu+2y-3=0</math> at points <math display="inline">M(1;1;1)</math> and <math display="inline">N(1;1;-2)</math>. |
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| − | # Find maxima and minima of a function subject to a constraint (or several constraints): |
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| − | ## <math display="inline">u=x^2y^3z^4</math>, <math display="inline">2x+3y+4z=18</math>, <math display="inline">x>0</math>, <math display="inline">y>0</math>, <math display="inline">z>0</math>; |
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| − | ## <math display="inline">u=x-y+2z</math>, <math display="inline">x^2+y^2+2z^2=16</math>; |
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| − | ## <math display="inline">u=\sum\limits_{i=1}^ka_ix_i^2</math>, <math display="inline">\sum\limits_{i=1}^kx_i=1</math>, <math display="inline">a_i>0</math>; |
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| − | |||
| − | === Typical questions for seminar classes (labs) within this section === |
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| − | |||
| − | # Let us consider <math display="inline">u(x;y)=\begin{cases}1,&x=y^2,\\0,&x\neq y^2.\end{cases}</math> Show that this function has a limit at the origin along any straight line that passes through it (and all these limits are equal to each other), yet this function does not have limit as <math display="inline">(x;y)\to(0;0)</math>. |
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| − | # Find the largest possible value of directional derivative at point <math display="inline">M(1;-2;-3)</math> of function <math display="inline">f=\ln xyz</math>. |
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| − | # Find maxima and minima of functions <math display="inline">u(x, y)</math> given implicitly by the equations: |
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| − | ## <math display="inline">x^2+y^2+u^2-4x-6y-4u+8=0</math>, <math display="inline">u>2</math>; |
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| − | ## <math display="inline">x^3-y^2+u^2-3x+4y+u-8=0</math>. |
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| − | # Find maxima and minima of functions subject to constraints: |
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| − | ## <math display="inline">u=xy^2</math>, <math display="inline">x+2y-1=0</math>; |
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| − | ## <math display="inline">u=xy+yz</math>, <math display="inline">x^2+y^2=2</math>, <math display="inline">y+z=2</math>, <math display="inline">y>0</math>. |
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| − | |||
| − | === Test questions for final assessment in this section === |
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| − | |||
| − | # Find all points where the differential of a function <math display="inline">f(x;y)=(5x+7y-25)e^{-x^2-xy-y^2}</math> is equal to zero. |
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| − | # Show that function <math display="inline">\varphi=f\left(\frac xy;x^2+y-z^2\right)</math> satisfies the equation <math display="inline">2xz\varphi_x+2yz\varphi_y+\left(2x^2+y\right)\varphi_z=0</math>. |
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| − | # Find maxima and minima of function <math display="inline">u=2x^2+12xy+y^2</math> under condition that <math display="inline">x^2+4y^2=25</math>. Find the maximum and minimum value of a function |
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| − | # <math display="inline">u=\left(y^2-x^2\right)e^{1-x^2+y^2}</math> on a domain given by inequality <math display="inline">x^2+y^2\leq4</math>; |
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| − | |||
| − | === Section 2 === |
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| − | |||
| − | ==== Section title: ==== |
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| − | |||
| − | Integration of Functions of Several Variables |
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| − | |||
| − | === Topics covered in this section: === |
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| − | |||
| − | * Z-test |
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| − | * Double integrals. Fubini’s theorem and iterated integrals |
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| − | * Substituting variables in double integrals. Polar coordinates |
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| − | * Triple integrals. Use of Fubini’s theorem |
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| − | * Spherical and cylindrical coordinates |
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| − | * Path integrals |
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| − | * Area of a surface |
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| − | * Surface integrals |
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| − | |||
| − | === What forms of evaluation were used to test students’ performance in this section? === |
||
| − | |||
| − | <div class="tabular"> |
||
| − | |||
| − | <span>|a|c|</span> & '''Yes/No'''<br /> |
||
| − | Development of individual parts of software product code & 0<br /> |
||
| − | Homework and group projects & 1<br /> |
||
| − | Midterm evaluation & 1<br /> |
||
| − | Testing (written or computer based) & 1<br /> |
||
| − | Reports & 0<br /> |
||
| − | Essays & 0<br /> |
||
| − | Oral polls & 0<br /> |
||
| − | Discussions & 0<br /> |
||
| − | |||
| − | |||
| − | |||
| − | </div> |
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| − | === Typical questions for ongoing performance evaluation within this section === |
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| − | |||
| − | # Represent double integrals below as an iterated integrals (or a sum of iterated integrals) with different orders of integration: <math display="inline">\iint\limits_Df(x;y)\,dx\,dy</math> where <math display="inline">D=\left\{(x;y)\left|x^2+y^2\leq9,\,x^2+(y+4)^2\geq25\right.\right\}</math>. |
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| − | # Represent integral <math display="inline">I=\displaystyle\iiint\limits_Df(x;y;z)\,dx\,dy\,dz</math> as iterated integrals with all possible (i.e. 6) orders of integration; <math display="inline">D</math> is bounded by <math display="inline">x=0</math>, <math display="inline">x=a</math>, <math display="inline">y=0</math>, <math display="inline">y=\sqrt{ax}</math>, <math display="inline">z=0</math>, <math display="inline">z=x+y</math>. |
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| − | # Find line integrals of a scalar fields <math display="inline">\displaystyle\int\limits_{\Gamma}(x+y)\,ds</math> where <math display="inline">\Gamma</math> is boundary of a triangle with vertices <math display="inline">(0;0)</math>, <math display="inline">(1;0)</math> and <math display="inline">(0;1)</math>. |
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| − | |||
| − | === Typical questions for seminar classes (labs) within this section === |
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| − | |||
| − | # Change order of integration in the iterated integral <math display="inline">\int\limits_0^{\sqrt2}dy\int\limits_y^{\sqrt{4-y^2}}f(x;y)\,dx</math>. |
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| − | # Find the volume of a solid given by <math display="inline">0\leq z\leq x^2</math>, <math display="inline">x+y\leq 5</math>, <math display="inline">x-2y\geq2</math>, <math display="inline">y\geq0</math>. |
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| − | # Change into polar coordinates and rewrite the integral as a single integral: <math display="inline">\displaystyle\iint\limits_Gf\left(\sqrt{x^2+y^2}\right)\,dx\,dy</math>, <math display="inline">G=\left\{(x;y)\left|x^2+y^2\leq x;\, x^2+y^2\leq y\right.\right\}</math>. |
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| − | # Having ascertained that integrand is an exact differential, calculate the integral along a piecewise smooth plain curve that starts at <math display="inline">A</math> and finishes at <math display="inline">B</math>: <math display="inline">\displaystyle\int\limits_{\Gamma}\left(x^4+4xy^3\right)\,dx +\left(6x^2y^2-5y^4\right)\,dy</math>, <math display="inline">A(-2;-1)</math>, <math display="inline">B(0;3)</math>; |
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| − | |||
| − | === Test questions for final assessment in this section === |
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| − | |||
| − | # Domain <math display="inline">G</math> is bounded by lines <math display="inline">y=2x</math>, <math display="inline">y=x</math> and <math display="inline">y=2</math>. Rewrite integral <math display="inline">\iint\limits_Gf(x)\,dx\,dy</math> as a single integral. |
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| − | # Represent the integral <math display="inline">\displaystyle\iint\limits_Gf(x;y)\,dx\,dy</math> as iterated integrals with different order of integration in polar coordinates if <math display="inline">G=\left\{(x;y)\left|a^2\leq x^2+y^2\leq 4a^2;\,|x|-y\geq0\right.\right\}</math>. |
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| − | # Find the integral making an appropriate substitution: <math display="inline">\displaystyle\iiint\limits_G\left(x^2-y^2\right)\left(z+x^2-y^2\right)\,dx\,dy\,dz</math>, <math display="inline">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\}</math>. |
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| − | # Use divergence theorem to find the following integrals <math display="inline">\displaystyle\iint\limits_S(1+2x)\,dy\,dz+(2x+3y)\,dz\,dx+(3y+4z)\,dx\,dy</math> where <math display="inline">S</math> is the outer surface of a tetrahedron <math display="inline">\frac xa+\frac yb+\frac zc\leq1</math>, <math display="inline">x\geq0</math>, <math display="inline">y\geq0</math>, <math display="inline">z\geq0</math>; |
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| − | |||
| − | === Section 3 === |
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| − | |||
| − | ==== Section title: ==== |
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| − | |||
| − | Uniform Convergence of Functional Series. Fourier Series |
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| − | |||
| − | ==== Topics covered in this section: ==== |
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| − | |||
| − | * Uniform and point wise convergence of functional series |
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| − | * Properties of uniformly convergent series |
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| − | * Fourier series. Sufficient conditions of convergence and uniform convergence |
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| − | * Bessel’s inequality and Parseval’s identity. |
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| − | |||
| − | === What forms of evaluation were used to test students’ performance in this section? === |
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| − | |||
| − | <div class="tabular"> |
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| − | |||
| − | <span>|a|c|</span> & '''Yes/No'''<br /> |
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| − | Development of individual parts of software product code & 0<br /> |
||
| − | Homework and group projects & 1<br /> |
||
| − | Midterm evaluation & 0<br /> |
||
| − | Testing (written or computer based) & 1<br /> |
||
| − | Reports & 0<br /> |
||
| − | Essays & 0<br /> |
||
| − | Oral polls & 0<br /> |
||
| − | Discussions & 0<br /> |
||
| − | |||
| − | |||
| − | |||
| − | </div> |
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| − | === Typical questions for ongoing performance evaluation within this section === |
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| − | |||
| − | # Find out whether the following functional series converges uniformly on the indicated intervals. Justify your answer. <math display="inline">\sum\limits_{n=1}^{\infty}e^{-n\left(x^2+2\sin x\right)}</math>, <math display="inline">\Delta_1=(0;1]</math>, <math display="inline">\Delta_2=[1;+\infty)</math>; |
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| − | # <math display="inline">\sum\limits_{n=1}^{\infty}\frac{\sqrt{nx^3}}{x^2+n^2}</math>, <math display="inline">\Delta_1=(0;1)</math>, <math display="inline">\Delta_2=(1;+\infty)</math> |
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| − | # Show that sequence <math display="inline">f_n(x)=nx\left(1-x\right)^n</math> converges non-uniformly on <math display="inline">[0;1]</math> to a continuous function <math display="inline">f(x)</math>, but <math display="inline">\lim\limits_{n\rightarrow+\infty}\int\limits_0^1f_n(x)\,dx=\lim\limits_{n\rightarrow+\infty}\int\limits_0^1f(x)\,dx</math>. |
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| − | # Decompose the following function determined on <math display="inline">[-\pi;\pi]</math> into Fourier series using the standard trigonometric system <math display="inline">\left.\left\{1;\cos kx;\sin kx\right\}\right|_{k=1}^{\infty}</math>. Draw the graph of the sum of Fourier series obtained. <math display="inline">f(x)=\begin{cases}1,\;0\leq x\leq\pi,\\0,\;-\pi\leq x<0.\end{cases}</math> |
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| − | # Prove that if for an absolutely integrable function <math display="inline">f(x)</math> on <math display="inline">[-\pi;\pi]</math> |
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| − | ## <math display="inline">f(x+\pi)=f(x)</math> then <math display="inline">a_{2k-1}=b_{2k-1}=0</math>, <math display="inline">k\in\mathbb{N}</math>; |
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| − | ## <math display="inline">f(x+\pi)=-f(x)</math> then <math display="inline">a_0=0</math>, <math display="inline">a_{2k}=b_{2k}=0</math>, <math display="inline">k\in\mathbb{N}</math>. |
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| − | |||
| − | ==== Typical questions for seminar classes (labs) within this section ==== |
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| − | |||
| − | # Show that sequence <math display="inline">f_n(x)=nx\left(1-x^2\right)^n</math> converges on <math display="inline">[0;1]</math> to a continuous function <math display="inline">f(x)</math>, and at that <math display="inline">\lim\limits_{n\rightarrow+\infty}\int\limits_0^1f_n(x)\,dx\neq\lim\limits_{n\rightarrow+\infty}\int\limits_0^1f(x)\,dx</math>. |
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| − | # Show that sequence <math display="inline">f_n(x)=x^3+\frac1n\sin\left(nx+\frac{n\pi}2\right)</math> converges uniformly on <math display="inline">\mathbb{R}</math>, but <math display="inline">\left(\lim\limits_{n\rightarrow+\infty}f_n(x)\right)'\neq\lim\limits_{n\rightarrow+\infty}f'_n(x)</math>. |
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| − | # Decompose <math display="inline">\cos\alpha x</math>, <math display="inline">\alpha\notin\mathbb{Z}</math> into Fourier series on <math display="inline">[-\pi;\pi]</math>. Using this decomposition prove that <math display="inline">\cot y=\frac1y+\sum\limits_{k=1}^{\infty}\frac{2y}{y^2-\pi^2k^2}</math>. |
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| − | # Function <math display="inline">f(x)</math> is absolutely integrable on <math display="inline">[0;\pi]</math>, and <math display="inline">f(\pi-x)=f(x)</math>. Prove that |
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| − | ## if it is decomposed into Fourier series of sines then <math display="inline">b_{2k}=0</math>, <math display="inline">k\in\mathbb{N}</math>; |
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| − | ## if it is decomposed into Fourier series of cosines then <math display="inline">a_{2k-1}=0</math>, <math display="inline">k\in\mathbb{N}</math>. |
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| − | # ## Decompose <math display="inline">f(x)=\begin{cases}1,\;|x|<\alpha,\\0,\; \alpha\leqslant|x|<\pi\end{cases}</math> into Fourier series using the standard trigonometric system. |
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| − | ## Using Parseval’s identity find <math display="inline">\sigma_1=\sum\limits_{k=1}^{\infty}\frac{\sin^2k\alpha}{k^2}</math> and <math display="inline">\sigma_2=\sum\limits_{k=1}^{\infty}\frac{\cos^2k\alpha}{k^2}</math>. |
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| − | |||
| − | === Test questions for final assessment in the course === |
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| − | |||
| − | # Find out whether the following functional series converge uniformly on the indicated intervals. Justify your answer. <math display="inline">\sum\limits_{n=1}^{\infty}\frac{xn+\sqrt n}{n+x}\ln\left(1+\frac x{n\sqrt n}\right)</math>, <math display="inline">\Delta_1=(0;1)</math>, <math display="inline">\Delta_2=(1;+\infty)</math>; |
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| − | # Show that sequence <math display="inline">f_n(x)=\frac{\sin nx}{\sqrt n}</math> converges uniformly on <math display="inline">\mathbb{R}</math> to a differentiable function <math display="inline">f(x)</math>, and at that <math display="inline">\lim\limits_{n\rightarrow+\infty}f'_n(0)\neq f'(0)</math>. |
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| − | |||
| − | === Section 1 === |
||
| − | |||
| − | ==== Section title: ==== |
||
| − | |||
| − | Integrals with Parameter(s) |
||
| − | |||
| − | === Topics covered in this section: === |
||
| − | |||
| − | * Definite integrals with parameters |
||
| − | * Improper integrals with parameters. Uniform convergence |
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| − | * 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? === |
||
| − | |||
| − | <div class="tabular"> |
||
| − | |||
| − | <span>|a|c|</span> & '''Yes/No'''<br /> |
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| − | Development of individual parts of software product code & 0<br /> |
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| − | Homework and group projects & 1<br /> |
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| − | Midterm evaluation & 0<br /> |
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| − | Testing (written or computer based) & 1<br /> |
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| − | Reports & 0<br /> |
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| − | Essays & 0<br /> |
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| − | Oral polls & 0<br /> |
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| − | Discussions & 0<br /> |
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| − | </div> |
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| − | === Typical questions for ongoing performance evaluation within this section === |
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| − | # Find out if <math display="inline">\displaystyle\int\limits_0^1\left(\lim\limits_{\alpha\to0}\frac{2x\alpha^2}{\left(\alpha^2+x^2\right)^2}\right)\,dx= \lim\limits_{\alpha\to0}\int\limits_0^1\frac{2x\alpha^2}{\left(\alpha^2+x^2\right)^2}\,dx</math>. |
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| − | # Differentiating the integrals with respect to parameter <math display="inline">\varphi</math>, find it: <math display="inline">I(\alpha)=\int\limits_0^{\pi/2}\ln\left(\alpha^2-\sin^2\varphi\right)\,d\varphi</math>, <math display="inline">\alpha>1</math>. |
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| − | # Prove that the following integral converges uniformly on the indicated set. <math display="inline">\displaystyle\int\limits_0^{+\infty}e^{-\alpha x}\cos2x\,dx</math>, <math display="inline">\Delta=[1;+\infty)</math>; |
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| − | # It is known that Dirichlet’s integral <math display="inline">\int\limits_0^{+\infty}\frac{\sin x}x\,dx</math> is equal to <math display="inline">\frac\pi2</math>. Find the values of the following integrals using Dirichlet’s integral |
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| − | ## <math display="inline">\int\limits_0^{+\infty}\frac\sin{\alpha x}x\,dx</math>, <math display="inline">\alpha\neq0</math>; |
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| − | ## <math display="inline">\int\limits_0^{+\infty}\frac{\sin x-x\cos x}{x^3}\,dx</math>. |
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| − | |||
| − | === Typical questions for seminar classes (labs) within this section === |
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| − | # Find out if <math display="inline">\displaystyle\int\limits_0^1\left(\int\limits_0^1f(x,\alpha)\,d\alpha\right)\,dx= \int\limits_0^1\left(\int\limits_0^1f(x,\alpha)\,dx\right)\,d\alpha</math> if <math display="inline">f(x;\alpha)=\frac{\alpha-x}{(\alpha+x)^3}</math>. |
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| − | # Find <math display="inline">\Phi'(\alpha)</math> if <math display="inline">\Phi(\alpha)=\int\limits_1^2\frac{e^{\alpha x^2}}x\,dx</math>. |
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| − | # Differentiating the integral with respect to parameter <math display="inline">\alpha</math>, find it: <math display="inline">I(\alpha)=\int\limits_0^\pi\frac1{\cos x} \ln\frac{1+\alpha\cos x}{1-\alpha\cos x}\,dx</math>, <math display="inline">|\alpha|<1</math>. |
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| − | # Find Fourier transform of the following functions: |
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| − | ## <math display="inline">f(x)=\begin{cases}1,&|x|\leq1,\\0,&|x|>1;\end{cases}</math> |
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| − | # Let <math display="inline">\widehat{f}(y)</math> be Fourier transform of <math display="inline">f(x)</math>. Prove that Fourier transform of <math display="inline">e^{i\alpha x}f(x)</math> is equal to <math display="inline">\widehat{f}(y-\alpha)</math>, <math display="inline">\alpha\in\mathbb{R}</math>. |
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| − | |||
| − | === Test questions for final assessment in this section === |
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| − | |||
| − | # Find out if <math display="inline">\displaystyle\int\limits_0^1\left(\int\limits_0^1f(x,\alpha)\,d\alpha\right)\,dx= \int\limits_0^1\left(\int\limits_0^1f(x,\alpha)\,dx\right)\,d\alpha</math> if <math display="inline">f(x;\alpha)=\frac{\alpha^2-x^2}{\left(\alpha^2+x^2\right)^2}</math>. |
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| − | # Find <math display="inline">\Phi'(\alpha)</math> if <math display="inline">\Phi(\alpha)=\int\limits_0^\alpha\frac{\ln(1+\alpha x)}x\,dx</math>. |
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| − | # Prove that the following integral converges uniformly on the indicated set. <math display="inline">\displaystyle\int\limits_{-\infty}^{+\infty}\frac{\cos\alpha x}{4+x^2}\,dx</math>, <math display="inline">\Delta=\mathbb{R}</math>; |
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| − | # Find Fourier integral for <math display="inline">f(x)=\begin{cases}1,&|x|\leq\tau,\\0,&|x|>\tau;\end{cases}</math> |
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