[Yu Yue education] reference materials of complex variable function and integral transformation of Northwestern Polytechnic University

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- Functions of complex variables and integral transformations - Chapter materials, examination materials - Northwest Polytechnic University 【】
Chapter I homework
Chapter one test
1、【 Single topic selection 】 The plural The sufficient and unnecessary condition for being a real number is
A、
B、
C、 It's a real number
D、 It's a real number
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2、【 Single topic selection 】 What is wrong in the following formula is
A、
B、
C、
D、
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3、【 Single topic selection 】 Known complex number , Then plural The corresponding principal value of the spokes
A、
B、（k For any integer ）
C、
D、(k For any integer )
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4、【 Single topic selection 】 The plural Trigonometric expression and exponential expression are
A、
B、
C、
D、
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5、【 Single topic selection 】 In the curve represented by the following equation , The circle is
A、
B、
C、
D、
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6、【 Single topic selection 】 The two ends of the hypotenuse of an isosceles right triangle are , Then the vertex of the right angle is
A、i or 1
B、1+i
C、2
D、1+i or 2
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7、【 Single topic selection 】 set up be yes
A、1
B、0
C、
D、
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8、【 Single topic selection 】 set up yes One of the first Cigen , be
A、100i
B、-100i
C、0
D、100
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9、【 Single topic selection 】 Which of the following is a multi connected region
A、
B、
C、
D、z
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10、【 Single topic selection 】 Which of the following is a bounded simply connected domain
A、
B、
C、
D、
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11、【 Single topic selection 】 The following is about functions of complex variables That's what I'm saying , The wrong is
A、 It corresponds to two functions of real variables
B、 It puts a little i Mapping to -i
C、 It is the z The angular area of the plane is mapped into w An angular area three times the size of the plane
D、 It has no inverse function
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12、【 Single topic selection 】 Which of the following functions is discontinuous
A、
B、
C、
D、
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13、【 Single topic selection 】
A、1
B、-1
C、0
D、i
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14、【 Single topic selection 】
A、0
B、
C、
D、 The limit does not exist
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15、【 Single topic selection 】 The following is about mapping , It's wrong to say
A、 This mapping will z Plane curve y=x mapping w Flat u=-v;
B、 This mapping will z Points on the plane mapping w Flat
C、 This mapping will z Plane curve x=1 mapping w Plane
D、 This mapping will z Plane curve mapping w Flat
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Chapter II homework
Chapter II test
1、【 Single topic selection 】 In area D The necessary and sufficient condition for inner derivability is
A、 stay D There is a certain point ,f(z) Resolve at this point
B、u,v stay D There is a partial derivative inside
C、u,v stay D I'm satisfied with C-R Conditions
D、f(z) stay D Analyze the inner part of
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2、【 Single topic selection 】 analytic function f(z)=u+iv The derivative of is
A、
B、
C、
D、
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3、【 Single topic selection 】 When a function is parsed at a certain point, it is derivable at that point
A、 Sufficient and unnecessary conditions
B、 Necessary but not sufficient conditions
C、 necessary and sufficient condition
D、 Neither necessary nor sufficient conditions
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4、【 Single topic selection 】 On the complex plane
A、 It's not continuous everywhere
B、 Everywhere continuous , You can't lead anywhere
C、 Everywhere continuous , Only in z=0 Place can lead
D、 Everywhere continuous , stay z=0 Processing analysis
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5、【 Single topic selection 】 The following statement is not true
A、 Functions of complex variables that are derivable at one point must be continuous
B、 Derivability and differentiability of functions of complex variables are equivalent
C、 The real part and imaginary part of complex variable function have first-order continuous partial derivative and must be analytically
D、 Functions of complex variables are derivable and analytically equivalent in the domain
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6、【 Single topic selection 】 The principal value of is
A、
B、
C、
D、
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7、【 Single topic selection 】 The following conclusion is incorrect
A、Lnz Is a multivalued function on the complex plane
B、 It's a periodic function
C、cosz Is an unbounded function
D、sinz It's a bounded function
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8、【 Single topic selection 】
A、2+2i
B、2-2i
C、-2+2i
D、-2-2i
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9、【 Single topic selection 】
A、-i
B、i
C、0
D、1
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10、【 Single topic selection 】 The correct one in the following formula is
A、
B、
C、
D、
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Chapter three homework
Chapter III test
1、【 Single topic selection 】 Set curve C From the origin to 1+2i Straight line segment of , be
A、
B、
C、
D、
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2、【 Single topic selection 】
A、0
B、1
C、2
D、3
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3、【 Single topic selection 】 curve C Is from the origin to 1+2i Straight line segment of , be The upper bound of is
A、
B、
C、5
D、
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4、【 Single topic selection 】
A、0
B、1
C、2
D、3
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5、【 Single topic selection 】
A、0
B、1
C、2
D、3
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6、【 Single topic selection 】 set up C For from 1 To i Any curve of , be
A、1
B、-1
C、2
D、0
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7、【 Single topic selection 】 Set curve C Starting from 1, The finish for 1+i, be The condition for its establishment is
A、f(z) stay C Continuous on
B、f(z) stay C Up to guide
C、f(z) stay C Upper resolution
D、f(z) stay C Upper differentiable
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8、【 Single topic selection 】
A、
B、
C、
D、
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9、【 Single topic selection 】 The following integrals that can be directly solved by Cauchy integral formula are
A、
B、
C、
D、
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10、【 Single topic selection 】
A、
B、
C、
D、
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11、【 Single topic selection 】 The following conclusion is correct
A、
B、
C、
D、
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12、【 Single topic selection 】 In the plural z Express
A、
B、
C、
D、
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13、【 Single topic selection 】 The conjugate harmonic function of is
A、
B、
C、
D、
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14、【 Single topic selection 】 The following statement is wrong
A、 Is a harmonic function
B、 Is a harmonic function
C、 It's an analytic function
D、 Not an analytic function
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15、【 Single topic selection 】 It is known that u=ky(x-1) Is a harmonic function , be k=
A、 Any real number
B、0
C、1
D、2
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16、【 Single topic selection 】 It is known that u=x+y, Is that u+iv Analytic v=
A、y+x
B、y-x
C、y-x+c
D、y+x+c
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Chapter IV Operation
Chapter IV test
1、【 Single topic selection 】 For plural columns The following statement is true
A、 convergence
B、 Divergence
C、 Equal to the limit 0
D、 Convergence and divergence cannot be determined
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2、【 Single topic selection 】 Which of the following plural columns is divergent
A、
B、
C、
D、
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3、【 Single topic selection 】 Series of complex terms
A、 Divergence
B、 Absolute convergence
C、 Conditional convergence
D、 Unable to judge
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4、【 Single topic selection 】 Series of complex terms
A、 Divergence
B、 Absolute convergence
C、 Conditional convergence
D、 Unable to judge
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5、【 Single topic selection 】 Which of the following complex series conditions converges ？
A、
B、
C、
D、
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6、【 Single topic selection 】 Power series Convergence and divergence on the convergence circle , The most accurate description is
A、 convergence
B、 Divergence
C、 There are convergence points , There are also divergent points
D、 Convergence and divergence cannot be judged
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7、【 Single topic selection 】 When p1 when , Power series The convergence and divergence on the convergence circle is
A、 Convergent
B、 Diverging
C、 There are convergence points , There are also divergent points
D、 Convergence and divergence cannot be determined
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8、【 Single topic selection 】 The radius of convergence is
A、
B、1
C、
D、0
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9、【 Single topic selection 】 The radius of convergence is
A、
B、
C、
D、
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10、【 Single topic selection 】 Series of complex terms The sum function in the unit circumference is
A、
B、
C、
D、
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11、【 Single topic selection 】 function In the torus domain Laurent series expansion of
A、
B、
C、
D、
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12、【 Single topic selection 】 function In the torus domain The Laurent expansion of is
A、
B、
C、
D、
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13、【 Single topic selection 】 function In the torus domain 0|z+1|2 Laurent is shown as
A、
B、
C、
D、
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14、【 Single topic selection 】 function arctanz stay z=0 Taylor's exhibition at is
A、
B、
C、
D、
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15、【 Single topic selection 】 function stay z=0 The power series in the neighborhood of is
A、
B、
C、
D、
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Chapter V homework
Chapter V test
1、【 Single topic selection 】 function The singularities of are
A、0
B、1
C、-1
D、 All of these are
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2、【 Single topic selection 】 Let's set the function f(z) In area Analyze the inner part of , be yes f(z) The necessary and sufficient condition for the singularity of nature is
A、 limit It's a complex constant
B、 limit
C、 limit Neither does it exist nor is it infinite
D、 None of the above
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3、【 Single topic selection 】z=0 Is the function What level of zero
A、1
B、3
C、5
D、7
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4、【 Single topic selection 】z=0 Is the function Several poles of ？
A、1
B、2
C、3
D、4
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5、【 Single topic selection 】z=0 Is the function Of
A、 Can go to the singularity
B、 First pole
C、 Natural singularity
D、 Secondary pole
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6、【 Single topic selection 】 function (n As a positive integer ) stay The residue at is
A、
B、
C、
D、
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7、【 Single topic selection 】
A、
B、
C、
D、
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8、【 Single topic selection 】
A、
B、
C、
D、
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9、【 Single topic selection 】 function stay The residue at is
A、
B、
C、
D、
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10、【 Single topic selection 】 It is known that , that
A、
B、
C、
D、
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11、【 Single topic selection 】 It is known that that
A、
B、
C、
D、
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12、【 Multiple choice 】 set up 0 yes f(z) Of m Level pole , seek f(z) stay 0 The residue at , The following is true
A、
B、
C、
D、
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13、【 Multiple choice 】 Calculate the form with residue, such as Integral , Which of the following is correct when replacing variables
A、
B、
C、
D、
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Chapter VI operation
Chapter 6 test
1、【 Single topic selection 】 function The Laplace transform of is
A、
B、
C、
D、
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2、【 Single topic selection 】 function The Laplace transform of is
A、
B、
C、
D、
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3、【 Single topic selection 】 function The Laplace transform of is
A、
B、
C、
D、
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4、【 Single topic selection 】 function The Laplace transform of is
A、
B、
C、
D、
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5、【 Single topic selection 】 function The Laplace transform of is
A、
B、
C、
D、
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6、【 Single topic selection 】 function The inverse Laplace transform of is
A、
B、
C、
D、
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7、【 Single topic selection 】 function The inverse Laplace transform of is
A、
B、
C、
D、
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8、【 Single topic selection 】 function The inverse Laplace transform of is
A、
B、
C、
D、
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9、【 Single topic selection 】 function and The convolution of is
A、t+sint
B、cost-sint
C、t-sint
D、t
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10、【 Single topic selection 】 Use Laplace transform to solve differential equations The solution of is
A、
B、
C、
D、
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Final exam of complex variable function and integral transformation
1、【 Single topic selection 】 The following is about the multiplication and quotient of complex numbers , The wrong is
A、 The module of the product of two complex numbers is equal to the product of their modules .
B、 The argument principal value of the product of two complex numbers is equal to the sum of their argument principal values .
C、 The module of two complex quotients is equal to the quotient of their module .
D、 The spokes of two complex quotients are equal to the difference between the spokes of the dividend and the divisor .
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2、【 Single topic selection 】 The following is about the powers and roots of complex numbers , What's not right is
A、n The product of the same complex number is called a power .
B、 The plural z Of n The second root has n Different values .
C、z All of the n The modules of secondary roots are the same .
D、z All of the n The spokes of the secondary root have n Different values .
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3、【 Single topic selection 】 Next, about neighborhood , Inside point , Concepts of open sets and regions , The wrong is
A、 The neighborhood of is an inequality The determined point set .
B、 The interior point must belong to a neighborhood .
C、 All points in an open set are interior points .
D、 A region is a connected open set .
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4、【 Single topic selection 】 The following is about functions of complex variables That's what I'm saying , The wrong is
A、 It corresponds to two functions of real variables
B、 It puts a little i Mapping to -1
C、 It is the z The angular domain of the plane is mapped to w An angular domain of the same size on the plane .
D、 It is the z Two families of hyperbolas on the plane are mapped into w Two families of parallel lines on the plane
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5、【 Single topic selection 】 On functions of complex variables The continuity of , The right is
A、 Functions are continuous at any point
B、 Functions are discontinuous at any point
C、 Continuous at the origin , Other points are discontinuous
D、 Discontinuous at the origin , Other points are continuous
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6、【 Single topic selection 】 set up , The following statement is true
A、f(z) stay z=0 Processing analysis
B、f(z) stay Processing analysis
C、f(z) stay z=0 The derivative at is 0
D、f(z) It is not differentiable everywhere in the plane
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7、【 Single topic selection 】( )
A、
B、
C、sin 1
D、cos 1
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8、【 Single topic selection 】 The following description is incorrect
A、cos z It's boundless
B、 It's a periodic function
C、 Negative logarithm is no longer true in complex numbers
D、 The power function is single valued
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9、【 Single topic selection 】 On the complex plane , In the following propositions , The wrong is
A、cos z It's a periodic function
B、 It's an analytic function
C、
D、
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10、【 Single topic selection 】 By default, the positive direction of the closed curve is counterclockwise , Set a closed curve C There are two disjoint closed curves , Then the compound closed circuit composed of three closed curves is
A、
B、
C、
D、
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11、【 Single topic selection 】 In the whole complex plane All the original functions of are
A、
B、
C、
D、
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12、【 Single topic selection 】 The following functions are not harmonic functions:
A、
B、
C、
D、
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13、【 Single topic selection 】 Complex column The limit of
A、1
B、-1
C、0
D、 non-existent
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14、【 Single topic selection 】 Complex column Is a series of complex terms Convergent
A、 sufficient condition
B、 Necessary condition
C、 necessary and sufficient condition
D、 It is neither sufficient nor necessary
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15、【 Single topic selection 】 If the power series stay z=2 Convergence at , So in z=1 It's about
A、 Divergence
B、 Absolute convergence
C、 Conditional convergence
D、 Undetermined
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16、【 Single topic selection 】 Power series The convergence radius of is equal to
A、0
B、1
C、
D、 non-existent
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17、【 Single topic selection 】 function The McLaughlin series of is
A、
B、
C、
D、
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18、【 Single topic selection 】 In the annular domain 0| z | The Laurent exhibition is
A、
B、
C、
D、
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19、【 Single topic selection 】z=0 yes Of
A、 Can go to the singularity
B、 First order singularity
C、 Secondary pole
D、 Third pole
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20、【 Single topic selection 】z=1 Is the function Of （ ） Level pole
A、1
B、2
C、3
D、4
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21、【 Single topic selection 】 The following statement is wrong
A、 Singularities are always isolated
B、 Isolated Singularities include removable singularities 、 Poles and natural singularities
C、 The Laurent expansion of a function at its natural singularity contains infinite negative power terms
D、 The analytic function must have no singularity in the analytic region
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22、【 Single topic selection 】 function stay The residue at is
A、2
B、-1
C、1
D、0
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23、【 Single topic selection 】 function stay The residue at is
A、2
B、i
C、1
D、0
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24、【 Single topic selection 】 Let's set the function , be Of Laplace Transformation is defined as
A、
B、
C、
D、
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25、【 Single topic selection 】f(t)=sin kt（k The real number ） Of Laplace Transformation for
A、
B、
C、
D、
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26、【 Single topic selection 】 The inverse Laplace transform of is
A、
B、
C、
D、
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27、【 Single topic selection 】 use Laplace Solving ordinary differential equations by transformation （ Group ） The step is （ ）（1） Take Laplace transform to transform the differential equation （ Group ） Algebraic equations transformed into image functions （ Group ）;（2） Solve the algebraic equation to get the image function ;（3） Take the inverse Laplace transform to get the differential equation （ Group ） Solution .
A、(2)(1)(3)
B、(1)(2)(3)
C、(3)(2)(1)
D、(1)(3)(2)
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28、【 Single topic selection 】Laplace Among the properties of transformation, the differential property of image function is
A、
B、
C、
D、
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29、【 Single topic selection 】 Unit step function Of Laplace Transformation for
A、
B、
C、
D、
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30、【 Single topic selection 】 Two plurals 2+i And 2-i The result of multiplication is
A、2
B、4-2i
C、5
D、3-2i
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31、【 Judgment questions 】 function f(z) In area D The inner derivative is in f(z) In area D The necessary and sufficient condition of inner parsing .
A、 correct
B、 error
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32、【 Judgment questions 】 The sum function of a power function is resolved in its convergence circle .
A、 correct
B、 error
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33、【 Judgment questions 】f(x,y)=u(x,y)+iv(x,y) In area D The necessary and sufficient condition for inner parsing is ：u(x,y) And v(x,y) In area D It's internally differentiable , And meet C-R equation .
A、 correct
B、 error
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34、【 Judgment questions 】z=0 Is the function The removable singularity .
A、 correct
B、 error
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35、【 Judgment questions 】 function stay z=0 It's about Taylor The radius of convergence of the expansion is 3.
A、 correct
B、 error
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36、【 Judgment questions 】 If , among C It is a positive closed curve of complex plane , be f(z) stay C The enclosed area must be analyzed .
A、 correct
B、 error
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37、【 Judgment questions 】 function f(z) stay The necessary and sufficient condition of point analysis is that it can be expanded into Power series of , And the expansion is the only .
A、 correct
B、 error
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38、【 Judgment questions 】 If the function stay Processing analysis , be Also in the Processing analysis .
A、 correct
B、 error
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39、【 Judgment questions 】 function It's a periodic function .
A、 correct
B、 error
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40、【 Judgment questions 】z=0 yes Of 4 Level pole .
A、 correct
B、 error
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