CA Part 2

Question 1

What is the definition of a smooth curve in the complex plane?

Answer 1

A curve is smooth if x(t), y(t) are continuously differentiable and z’(t) ≠ 0.

Question 2

How is the line integral of a function f(z) along a curve C defined?

Answer 2

The line integral is defined as lim (n → ∞) Σ f(zk) Δzk.

Question 3

What is the effect of reversing the direction of integration along a curve?

Answer 3

Reversing the direction introduces a minus sign.

Question 4

What is a contour integral?

Answer 4

A contour integral is an integral over a closed curve and is written as I f(z) dz.

Question 5

What is the Cauchy theorem?

Answer 5

If f(z) is analytic everywhere within and on a simple closed curve C, then I f(z) dz = 0.

Question 6

What is the significance of Green’s theorem in complex analysis?

Answer 6

Green’s theorem relates line integrals around a simple closed curve to double integrals over the region it encloses.

Question 7

What does the Morera theorem state?

Answer 7

If f(z) is continuous in a region R and I f(z) dz = 0 for every closed curve C in R, then f(z) is analytic in R.

Question 8

What does path independence imply in complex analysis?

Answer 8

The line integral of f(z) between two points P and Q is the same for any two paths from P to Q if f(z) is analytic in the region.

Question 9

What is contour deformation?

Answer 9

If f(z) is analytic in and on the boundary between two simple closed curves C1 and C2, then I f(z) dz = I f(z) dz.

Question 10

What is the Darboux inequality?

Answer 10

The Darboux inequality states that |I f(z) dz| ≤ M L, where M is an upper bound on |f(z)| on C and L is the length of C.

Question 11

What is the Cauchy integral formula?

Answer 11

If f(z) is analytic inside and on a simple closed curve C and z0 is inside C, then f(z0) = (1/2πi) I (f(z)/(z-z0)) dz.

Question 12

What happens to the integral of an analytic function over a closed contour?

Answer 12

The integral is zero, as stated by the Cauchy theorem.

Question 13

Fill in the blank: The integral of f(z) dz over a closed curve can be expressed as I f(z) dz = I f(z) dz.

Answer 13

C1 C2

Question 14

True or False: The Cauchy integral formula can be used to find derivatives of analytic functions.

Answer 14

True.

Question 15

What is a consequence of Cauchy’s theorem regarding closed paths?

Answer 15

The line integral along any closed path in a region where f(z) is analytic is zero.

Question 16

What is the result of integrating a function that is analytic on and within a contour?

Answer 16

The integral is independent of the path taken between two points.

Question 17

What is the relationship between the Cauchy integral formula and the value of f(z) at a point inside the contour?

Answer 17

The formula gives the value of f(z) at a point inside the contour in terms of an integral around the contour.

Question 18

What is the Cauchy integral formula for derivatives?

Answer 18

If f(z) is analytic everywhere within and on a simple closed curve C and z0 is any point inside C, then
f(n)(z0) = 2 i IC (f(z)/(z - z0)^(n + 1)) dz/n!

This formula allows for the calculation of derivatives of analytic functions using contour integrals.

Question 19

What does the Cauchy inequality state about the derivatives of analytic functions?

Answer 19

If f(z) is analytic within and on a circle C of radius R with center at z0, and |f(z)| ≤ M on C, then |f(n)(z0)| ≤ M/(nR^n)

This inequality provides an upper bound on the magnitude of derivatives of analytic functions based on their maximum value on a contour.

Question 20

Fill in the blank: The maximum radius, R, about z0 for which the Taylor series converges is called the ______.

Answer 20

radius of convergence

Question 21

What is Liouville’s theorem?

Answer 21

If f(z) is bounded and analytic for all values of z, then f(z) is constant.

This theorem implies that entire functions that are bounded must be constant functions.

Question 22

What is the Fundamental Theorem of Algebra?

Answer 22

A non-constant polynomial has at least one zero.

This theorem establishes that every polynomial function of degree n has exactly n roots in the complex plane, counting multiplicities.

Question 23

What is a Laurent series?

Answer 23

A generalization of Taylor series for functions with singularities, allowing representation in an annulus centered at z0.

Laurent series can include negative powers of (z - z0) and are useful in complex analysis for functions that are not analytic everywhere.

Question 24

What does the Cauchy integral formula imply about the derivatives of an analytic function?

Answer 24

The derivatives of f(z) at z0 of all orders exist and are analytic.

This is a fundamental property of analytic functions, indicating their smoothness and differentiability.

Question 25

True or False: The series S(z) = ∑ cn(z - z0)^n converges for some point w ≠ z0 implies that S(z) is an analytic function of z.

Answer 25

True

Question 26

What is the relationship between singularities and the radius of convergence for a Taylor series?

Answer 26

If f(z) has singularities, then R extends to the nearest singularity. ## Footnote The radius of convergence is determined by the distance to the closest singularity from the point of expansion.

Question 27

Fill in the blank: The ratio test shows that the Maclaurin series for f(z) = (1 - z)^2 is absolutely convergent for all |z| < ______.

Answer 27

1

Question 28

What does Taylor’s theorem state about analytic functions?

Answer 28

If f(z) is analytic for all z such that |z - z0| < R0, then f(z) can be expressed as a power series around z0. ## Footnote This theorem allows for the approximation of analytic functions using polynomials.

Question 29

What is the significance of the Cauchy integral formula in complex analysis?

Answer 29

It provides a method for computing values of analytic functions and their derivatives using contour integrals. ## Footnote This formula is foundational in the study of complex functions and their properties.

Question 30

What is the general form of the Laurent series for a function f(z) around a singular point z0?

Answer 30

f(z) = Σ an (z - z0)^n ## Footnote This series is valid in a neighborhood around z0 until the next singular point.

Question 31

What characterizes an isolated singular point of a function f(z)?

Answer 31

f(z) is analytic in a neighborhood surrounding the point ## Footnote An isolated singular point allows the function to have a Laurent series expansion around it.

Question 32

What is a pole of order m at a singular point z0?

Answer 32

The negative powers in the Laurent series are finite in number. ## Footnote A pole of order 1 is a simple pole, while a pole of order 2 is a double pole.

Question 33

What is an essential singularity?

Answer 33

The number of negative powers in the Laurent series is infinite. ## Footnote This type of singularity does not allow for a finite expansion in terms of negative powers.

Question 34

What is the residue of a function f(z) at a singular point z0?

Answer 34

The coefficient a1 in the Laurent series expansion of f(z). ## Footnote The residue can be calculated using the residue formula.

Question 35

What does the Cauchy residue theorem state?

Answer 35

I f(z) dz = 2 i * (sum of residues inside C) ## Footnote This theorem applies to functions that are analytic everywhere except at a finite number of singularities within a closed contour.

Question 36

How do you evaluate the residue of a function with a simple pole at z0?

Answer 36

a1 = lim (z → z0) [(z - z0) f(z)] ## Footnote This limit gives the residue directly for simple poles.

Question 37

What is the method to find the residue for a pole of order 2?

Answer 37

Differentiate (z - z0)^2 f(z) and evaluate the limit at z0. ## Footnote This process removes the leading terms, allowing for the calculation of the residue.

Question 38

What happens if f(z) has a removable singularity at z0?

Answer 38

f(z) can be defined at z0 to make it analytic. ## Footnote Setting f(0) = lim (z → 0) f(z) can remove the singularity.

Question 39

What is the significance of the term g(z)/(z - z0)^m in determining the order of a pole?

Answer 39

g(z) is analytic at z0 and g(z0) ≠ 0. ## Footnote This indicates that f(z) has a pole of order m at z0.

Question 40

True or False: A function can have multiple singular points.

Answer 40

True ## Footnote The residue theorem can be applied to each singularity within a contour.

Question 41

Fill in the blank: The integral of a function f(z) around a closed contour C that does not enclose any singularities is ______.

Answer 41

0 ## Footnote This is a result of Cauchy's integral theorem.

Question 42

What is the formula for the contour integral involving a function with multiple singularities?

Answer 42

I f(z) dz = 2 i (r1 + r2 + ...) ## Footnote Each ri corresponds to the residue at each singular point zi.

Question 43

What is the process for expanding g(z) when determining the nature of a singularity?

Answer 43

Expand g(z) in a Taylor series around z0. ## Footnote This helps identify the order of the singularity based on the coefficients.

Question 44

What is one of the beautiful applications of Complex Analysis?

Answer 44

The evaluation of classes of real integrals ## Footnote This refers to the use of complex techniques to solve integrals that are challenging in real analysis.

Question 45

What type of integrals are considered in the text?

Answer 45

Integrals of the form I = ∫ F(sin θ, cos θ) dθ from 0 to 2π ## Footnote F is a rational function that is non-singular for θ in the interval [0, 2π].

Question 46

What transformation is used to evaluate the integral I?

Answer 46

Convert I to a contour integral and evaluate it using the residue theorem ## Footnote This involves changing variables to express the integral in terms of complex exponentials.

Question 47

What is the substitution made for z in the contour integral?

Answer 47

z = exp(iθ) ## Footnote This substitution maps the angle θ to the unit circle in the complex plane.

Question 48

What contour is used for the evaluation of the integral?

Answer 48

The unit circle centered at the origin ## Footnote The contour integral is evaluated by integrating around this unit circle.

Question 49

What does F(sin θ, cos θ) become in terms of z?

Answer 49

F(sin θ, cos θ) = F(z/(1+z), (1-z)/(1+z)) ## Footnote This transformation expresses the sine and cosine functions in terms of the complex variable z.

Question 50

What theorem is applied to find the integral I?

Answer 50

The Cauchy residue theorem ## Footnote This theorem allows for the evaluation of integrals based on the residues of singular points inside the contour.

Question 51

What is the archetypal example of a real integral discussed?

Answer 51

I = ∫ (a + b cos θ) dθ from 0 to 2π with a > |b| > 0 ## Footnote This specific integral serves as a classic example in the evaluation of integrals using complex analysis.

Question 52

What is the result of the integral I = ∫ (a + b cos θ) dθ?

Answer 52

I = 2π (a) ## Footnote This result is derived using the residue theorem and the properties of the integrand.

Question 53

What are the conditions for integrals of the form ∫ P(x) dx / Q(x)?

Answer 53

Q(x) ≠ 0 for all real x and the degree of Q(x) is at least 2 greater than the degree of P(x) ## Footnote These conditions ensure that the integral can be evaluated using residues.

Question 54

What is the formula for the integral involving P(x) and Q(x)?

Answer 54

∫ P(x) dx / Q(x) = 2i Σ residues of P(z)/Q(z) in upper half-plane ## Footnote This relates to the evaluation of real integrals through complex analysis.

Question 55

What is the result of the integral I = ∫ (1 + x^2)^(-3) dx?

Answer 55

I = 2i r = 38 ## Footnote This is derived from finding the residue at the pole in the upper half-plane.

Question 56

What types of integrals allow for the relaxation of conditions if poles are of order 1?

Answer 56

Types II and III integrals ## Footnote This means that even if there are poles on the real axis, certain techniques can still be applied.

Question 57

What is the general form of the integral for type III?

Answer 57

∫ exp(i x) P(x) dx / Q(x) ## Footnote This form requires specific conditions regarding the degrees of P(x) and Q(x).

Question 58

What is the role of the residue theorem in the evaluation of integrals?

Answer 58

It allows for the calculation of integrals based on residues of poles within the contour ## Footnote The residue theorem is fundamental in complex analysis for evaluating integrals over closed curves.

Question 59

What is the condition for f(z) in the general case of integrals?

Answer 59

f(z) is analytic except at most at a finite number of isolated points ## Footnote This ensures that the function behaves well within the contour of integration.

Question 60

What is the formula for IC?

Answer 60

IC = ZCR + Z ## Footnote This represents a mathematical relationship involving integrals and limits.

Question 61

What happens to the integral over CR as R approaches 1?

Answer 61

The integral over CR again vanishes as R ! 1 ## Footnote This is under the remaining conditions for integrals of type II and III.

Question 62

What is the result of the limit of the integral of f(z) with a simple pole on the real axis?

Answer 62

lim ZC f(z) dz = i (residue of f(z) on real axis) ## Footnote This indicates the connection between residues and integrals in complex analysis.

Question 63

What does p.v. denote in the context of integrals?

Answer 63

p.v. denotes the principal value of the integral ## Footnote Principal value is used when dealing with singularities in integrals.

Question 64

What is the relationship between the principal value of an integral and residues in the upper half-plane?

Answer 64

1 p.v. f(x) dx = 2i X residues of f(z) in upper half-plane + i X residues of f(z) on real axis ## Footnote This highlights the importance of residues in evaluating certain integrals.

Question 65

What is the integral I for the function sin(x)/x?

Answer 65

I = Z 1 sin(x) / x dx ## Footnote This is a common integral in complex analysis.

Question 66

What is the residue r for f(z) = exp(iz)/z at z = 0?

Answer 66

r = lim (z -> 0) z exp(iz) = 1 ## Footnote The residue is crucial for evaluating the integral involving this function.

Question 67

What is the value of p.v. Z 1 cos(x)/x dx?

Answer 67

p.v. Z 1 cos(x)/x dx = 0 ## Footnote This is due to the integrand being an odd function.

Question 68

What is the value of p.v. Z 1 sin(x)/x dx?

Answer 68

p.v. Z 1 sin(x)/x dx = i ## Footnote The sine integral has a removable singularity at x = 0.

Question 69

What integral is represented by I = Z 1 sin^2(x)/x^2 dx?

Answer 69

I = Z 1 sin^2(x)/x^2 dx ## Footnote This integral is a specific case often analyzed in the context of Fourier analysis.

Question 70

What is the residue for f(z) = exp(2iz)/z^2 at z = 0?

Answer 70

r = lim (z -> 0) (1/exp(2iz))/z^2 = 2i ## Footnote This residue is used in evaluating the integral involving sin^2(x).

Question 71

What is the result of the integral Z 1 cos(2x) dx / x^2?

Answer 71

Z 1 cos(2x) dx / x^2 = 2 ## Footnote This is derived from the calculation involving the residue.

Question 72

What is the relationship between the two examples provided?

Answer 72

The two examples are related through their integrands and the use of residues ## Footnote Both integrals involve sine functions and share similar methods of evaluation.