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mathematics
calculus

Questions and Answers of Calculus

By substituting 1/(1 ˆ’ z) for z in the expansionFound in Exercise 1, derive the Laurent series representation (Compare with Example 2, Sec. 65.)
Find the Taylor series for the functionAbout the point z0 = 2. Then, by differentiating that series term by term, show that
With the aid of series, show that the function ( defined by means of the equationsis entire. Use that result to establish the limit
In the w plane, integrate the Taylor series expansion (see Example 4, Sec. 59)Along a contour interior to the circle of convergence from w = 1 to w = z to obtain the representation
Use multiplication of series to show that
Use the expansionin Example 2, Sec. 67, and the method illustrated in Example 1, Sec. 62, to show that When C is the positively oriented unit circle |z| = 1.
Use mathematical induction to establish Leibniz' rule (Sec. 67)For the nth derivative of the product of two differentiable functions ((z) and g(z). The rule is valid when n = 1. Then, assuming that
Let ( (z) be an entire function that is represented by a series of the form(a) By differentiating the composite function g(z) = ([( (z)] successively, find the first three nonzero terms in the
The Euler numbers are the numbers En (n = 0, 1, 2( ( ( () in the Maclaurin series representationPoint out why this representation is valid in the indicated disk and why Then show that
Find the residue at z = 0 of the function (a) 1 / z + z2; (b) z cos (1/z); (c) z - sin z / z; (d) cot z / z4; (e) sinh z / z4 (1 - z2)
Use the theorem in Sec. 71, involving a single residue, to evaluate the integral of each of these functions around the circle |z| = 2 in the positive sense: (a) z5 / 1 - z3; (b) 1 / 1 + z2; (c) 1 / z.
Let C denote the circle |z| = 1, taken counterclockwise, and use the following steps to show that(a) By using the Maclaurin series for ez and referring to Theorem 1 in Sec. 65, which justifies the
Let the degrees of the polynomialsP(z) = a0 + a1z + a2z2 +· · ·+ anzn (an ‰ 0)AndQ(z) = b0 + b1z + b2z2 +· · ·+ bmzm (bm ‰ 0)Be such
In each case, write the principal part of the function at its isolated singular point and determine whether that point is a pole, a removable singular point, or an essential singular
In each case, write the principal part of the function at its isolated singular point and determine whether that point is a pole, a removable singular point, or an essential singular point:(a)(b) (c)
Suppose that a function f is analytic at z0, and write g(z) = f (z)/(z − z0). Show that (a) If f(z0) ≠ 0, then z0 is a simple pole of g, with residue f (z0); (b) If f(z0) = 0, then z0 is a
Write the functionAs Point out why Ï†(z) has a Taylor series representation about z = ai, and then use it to show that the principal part of f at that point is
In each case, show that any singular point of the function is a pole. Determine the order m of each pole, and find the corresponding residue B. (a) z2 + 2 / z - 1; (b) (z / 2z + 1)3; (c) exp z / z2 +
Show that(a)(b) (c)
Find the value of the integralTaken counterclockwise around the circle (a) |z ˆ’ 2| = 2; (b) |z| = 4
Find the value of the integralTaken counterclockwise around the circle (a) |z| = 2; (b) |z + 2| = 3
Evaluate the integralWhen C is the circle |z| = 2, described in the positive sense.
Use the theorem in Sec. 71, involving a single residue, to evaluate the integral of f (z) around the positively oriented circle |z| = 3 when(a)(b) (c)
Show that(a)(b)
Show that(a)Where zn = Ï€/2 + nÏ€ (n = 0, ±1, ±2, . . .); (b) Where zn = (Ï€/2 + nÏ€) i (n = 0, ±1, ±2, . . .);
Let C denote the positively oriented circle |z| = 2 and evaluate the integral(a)(b)
Let CN denote the positively oriented boundary of the square whose edges lie along the linesWhere N is a positive integer, Show that Then, using the fact that the value of this integral tends to zero
Show thatWhere C is the positively oriented boundary of the rectangle whose sides lie along the lines x = ±2, y = 0, and y = 1. Suggestion: By observing that the four zeros of the polynomial
Consider the functionWhere q is analytic at z0, q(z0) = 0, and q'(z0) ‰ 0. Show that z0 is a pole of order m = 2 of the function f, with residue Suggestion: that z0 is a zero of order m =
Use the result in Exercise 7 to find the residue at z = 0 of the function (a) f(z) = csc2z; (b) f(z) = 1/(z + z2)2
Use residues to evaluate the improper integrals in Exercises
Use residues to find the Cauchy principal values of the integrals in Exercises
Use a residue and the contour shown in Fig. 95, where R > 1, to establish the integration formula
Let m and n be integers, where 0 ‰¤ m(a) Show that the zeros of the polynomial z2n + 1 lying above the real axis are And that there are none on that axis (b) With the aid of Theorem 2 in
Use residues to evaluate the improper integrals in Exercises
Use residues to find the Cauchy principal values of the improper integrals in Exercises
Follow the steps below to evaluate the Fresnel integrals, which are important in diffraction theory:(a) By integrating the function exp(iz2) around the positively oriented boundary of the sector 0
In Exercises, take the indented contour in Fig. 101 (Sec. 82).Use the function to show that
Use the functionto show that
Use residues to evaluate the definite integrals in Exercises
Suppose that a function f is analytic inside and on a positively oriented simple closed contour C and that it has no zeros on C. Show that if f has n zeros zk (k = 1, 2, . . . , n) inside C, where
Determine the number of zeros, counting multiplicities, of the polynomial (a) z6 − 5z4 + z3 − 2z ; (b) 2z4 − 2z3 + 2z2 − 2z + 9 Inside the circle |z| = 1.
Determine the number of zeros, counting multiplicities, of the polynomial (a) z4 + 3z3 + 6; (b) z4 − 2z3 + 9z2 + z − 1; (c) z5 + 3z3 + z2 + 1 Inside the circle |z| = 2.
Simplify as much as possible. Be sure to remove all parentheses and reduce all fractions. 1. 4 - 2(8 - 11) + 6 2. 3[2 - 4(7 - 12)] 3. -4[5 (-3 + 12 - 4) + 2 (13 - 7)]
Perform the indicated operations and simplify. 1. (3x - 4) (x + 1) 2. (2x - 3)2 3. (3x - 9) (2x + 1)
Find the value of each of the following; if undefined, say so. (a) (0 ∙ 0) (b) 0 / 0 (c) 0 / 17 (d) 3 / 0 (e) 05 (f) 170
Change rational number to a decimal by performing long division. 1. 1/12 2. 2/7 3. 3/21 4. 5/17 5. 11/3 6. 11/13
Change each repeating decimal to a ratio of two integers (see Example 1). 1. 0.123123123... 2. 0.217171717... 3. 0.56565656... 4. 3.929292... 5. 0.199999... 6. 0.399999...
Show that any rational number p/q, for which the prime factorization of q consists entirely of 2s and 5s, has s terminating decimal expansion.
Show that between any two different real numbers there is a rational number. Show that between any two different real numbers there are infinitely many rational numbers.
About how many times has your heart beat by your twentieth birthday?
The General Sherman tree in California is about 270 feet tall and averages about 16 feet in diameter. Estimate the number of board feet (1 board foot equals 1 inch by 12 inches by 12 inches) of
Write the convene and the contrapositive to the following statements. (a) If it rains today, then I will stay home from work. (b) If the candidate meets all the qualifications, then she will be hired.
Write the converse and the contrapositive to the following statements. (a) If I get an A on the final exam, I will pass the course. (b) If I finish my research paper by Friday, then I will take off
Write the converse and the contrapositive to the following statements. (a) (Let a, b, and c be the lengths of sides of a triangle.) If a2 b2 = c2, then the triangle is a right triangle. (b) If angle
Write the converse and the contrapositive to the following statements. (a) If the measure of angle ABC is 45°, then angle ABC is an acute angle. (b) If a < b then a2 < b2.
Use the rules regarding the negation of statements involving quantifiers to write the negation of the following statements. Which is true, the original statement or its negation? (a) Every isosceles
Use the rules regarding the negation of statements involving quantifiers to write the negation of the following statements. Which is true, the original statement or its negation? (a) Every natural
Which of the following are true? Assume that x and y are real numbers. (a) For every x, x > 0 ⇒ x2 > 0. (b) For every x, x > 0 ⇔ x2 > 0. (c) For every x, there exist a u such that y > x2. (d) For
Which of the following are true? Unless it is stated other-wise, assume that x, y, and e are real numbers. (a) For every x, x < x + 1. (b) There exists a natural number N such that all prime numbers
Prove the following statements. (a) If n is odd, then n2 is odd. (b) If n2 is odd, then n is odd.
According to the Fundamental Theorem of Arithmetic, every natural number greater than 1 can be written as the product of primes in a unique way, except for the order of the factors. For example, 45 =
Use the Fundamental Theorem of Arithmetic (Problem 75) to show that the square of any natural number greater than 1 can be written as the product of primes in a unique way, except for the order of
Show that √2 is irrational. Try a proof by contradiction. Suppose that √2 = p/q, where p and q are natural numbers (necessarily different from 1). Then 2 = p2/q2, and so 2q2 = p2. Now use problem
Show that √3 is irrational (see problem 77). In problem Show that √2 is irrational. Try a proof by contradiction. Suppose that √2 = p/q, where p and q are natural numbers (necessarily different
Show that the sum of two rational number is rational.
Shot that the product of a rational number (other than 0) and an irrational number is irrational.
Which of the following are rational and which are irrational? (a) - √9 (b) 0.375 (c) (3 √2) (5 √2) (d) (1 + √3)2
A number b is called an upper bound for a set S of numbers if x s b for all x in S. For example 5, 6.5, and 13 are upper bounds for the set S = {1,2,3, 4, 5}. The number 5 is the least upper bound
The Axiom of Completeness for the real numbers says: Every set of real numbers that has an upper bound has a least upper bound that is a real number. (a) Show that the italicized statement is false
Show each of the following intervals on the real line. (a) [-1, 1] (b) (-4, 1] (c) (-4, 1) (d) [1, 4] (e) [- 1, ∞) (f) (- ∞, 0]
Assume that a > 0, b > 0. Prove each statement. (a) a < b ⇔ a2 < b2 (b) a < b ⇔ 1/a > 1/b
In each of Problems 1-5, express the solution set of the given inequality in interval notation and sketch its graph. 1. 3 - 7 < 2x - 5 2. 3x - 5 < 4x - 6 3. 7x - 2 ≤ 9x + 3 4. 5x - 3 > 6x - 4 5. -4
Find all values of x that satisfy both inequalities simultaneously. (a) 3x + 7 > 1 and 2x + 1 < 3 (b) 3x + 7 > 1 and 2x + 1> -4 (c) 3x + 7 > 1 and 2x 1 < -4
Find all the values of x that satisfy at least one of the two inequalities. (a) 2x - 7 > 1 or 2x + 1 < 3 (b) 2x - 7 ≤ 1 or 2x + 1 < 3 (c) 2x - 7 ≤ 1 or 2x + 1 > 3
Solve each inequality. Express your solution in interval notation. (a) 1.99 < 1/ x < 2.01 (b) 2.99 < 1/x + 2 < 3.01
In problem 1-3, find the solution set of the given inequalities. 1. |x - 2| ≥ 5 2. |x + 2| < 1 3. |4x + 5| ≤ 10
In problem 1 - 3 solve the given quadratic inequality using the Quadratic Formula. 1. x2 - 3x - 4x ≥ 0 2. x2 - 4x + 4 ≤ 0 3. 3x2 + 17x - 6 > 0
In problem 1-3, show that the indicated implication is true. 1. |x - 3| < 0.5 ⇒ |5x - 15| < 2.5 2. |x + 2| < 0.3 ⇒ |4x + 8| < 1.2 3. |x - 2| < ε/6 ⇒ |6x - 12| < ε
In problem 1-3, find δ (depending on ε) so that the given implication is true. 1. |x - 5| < δ ⇒ |3x - 15| < ε 2. |x - 2| < δ ⇒ |4x - 8| < ε 3. |x + 6| < δ ⇒ |6x + 36| < ε
On a lathe, you are to turn out a disk (thin right circular cylinder) of circumference 10 inches. This is done by continually measuring the diameter as you make the disk smaller. How closely must you
Fahrenheit temperatures and Celsius temperatures are related by the formula C = 5/9(F - 32). An experiment requires that a solution be kept at 50°C with an error of at most 3% (or 1.5°). You have
Use the properties of the absolute value to show that each of the following is true. (a) |a - b| ≤ |a| + |b| (b) |a - b| ≥ |a| - |b| (c) |a + b + c| ≤ |a| + |b| + |c|
Use the Triangle Inequality and the fact that 0
Show each of the following: (a) x < x2 for x < 0 or x > 1 (b) x2 < x for 0 < x < 1
Show that a ≠ 0 ⇒ a2 + 1 / a2 ≥ 2.
Show that, among all rectangles with given perimeter p, the square has the largest are. If a and b denote the lengths of adjacent sides of a rectangle of perimeter p, then the area is ab, and for
The formula 1 / R = R / R1 + 1 / R2 + 1/ R3 gives the total resistance R in an electric circuit due to three resistances, R1, R2, and R3, connected in parallel. If 10 ≤ R1 ≤ 20, 20 ≤ R2 ≤ 30,
The radius of a sphere is measured to be about 10 inches. Determine a tolerance S in this measurement that will ensure an error of less than 0.01 square inch in the calculated value of the surface
In problem 1-4, plot the given points in the coordinate plane and then find the distance between them. 1. (3, 1), (1, 1) 2. (-3, 5), (2, - 2) 3. (4, 5), (5, - 8) 4. (- 1, 5), (6, 3)
Find the length of the segment joining the midpoints of the segments AB and CD, where A = (1, 3), B = (2, 6), C = (4, 7), and D = (3, 4).
In problems 1-3 find the equation of the circle satisfying the given conditions. 1. Center (1, 1) radius 1 2. Center (-2, 3), radius 4 3. Center (2, - 1), goes through (5, 3)
In problem 1-2 find the center and radius of the circle with the given equation. 1. x2 + 2x + 10 + y2 - 6y - 10 = 0 2. x2 + y2 - 6y = 16
In problem 1-6 find the slope of the line containing the given two points. 1. (1, 1) and (2, 2) 2. (3, 5) and (4, 7) 3. (2, 3) and (- 5, -6) 4. (2, - 4) and (0, - 6) 5. (3, 0) and (0, 5) 6. (- 6, 0)
In problem 1-4 find the slope and y-intercept of each line. 1. 3y = - 2x + 1 2. - 4y = 5x - 6 3. 6 - 2y = 10x - 2
Write an equation for the line through (3, - 3) that is (a) Parallel to the line y = 2x + 5; (b) Perpendicular to the line y = 2x + 5; (c) Parallel to the line 2x + 3y = 6; (d) Perpendicular to the
Find the value of c for which the line 3x + cy = 5 (a) Passes through the point (3, 1); (b) Is parallel to the y-axis; (c) is parallel to the line 2x + y = -1; (d) Has equal x- and y-intercepts; (e)
Find the value of k such that the line kx - 3y = 10 (a) Is parallel to the line y = 2x + 4; (b) Is perpendicular to the line y = 2x + 4; (c) Is perpendicular to the line 2x + 3y = 6.
Show that the equation of the line with x-intercept a ≠ 0 and y-intercept b ≠ 0 can be written as x /a + y / b = 1
In problem 1-3 find the coordinates of the point of intersection. Then write an equation for the line through that point perpendicular to the line given first. 1. 2x + 3y = 4 -3x + y = 5 2. 4x - 5y =

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