Mathematical Methods in the Physical Sciences
3rd Edition
ISBN: 9780471198260
Author: Mary L. Boas
Publisher: Wiley, John & Sons, Incorporated
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Chapter 14.1, Problem 18P
Find the real and imaginary parts
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Chapter 14 Solutions
Mathematical Methods in the Physical Sciences
Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...
Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.1 - Find the real and imaginary parts u(x,y) and...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21 . Use the Cauchy-Riemann conditions to...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - 1 to 21. Use the Cauchy-Riemann conditions to find...Ch. 14.2 - Using the definition (2.1) of (d/dz)f(z), show...Ch. 14.2 - Using the definition (2.1) of (d/dz)f(z), show...Ch. 14.2 - Prob. 27PCh. 14.2 - Using the definition (2.1) of (d/dz)f(z), show...Ch. 14.2 - Problem 28 is the chain rule for the derivative of...Ch. 14.2 - Problem 28 is the chain rule for the derivative of...Ch. 14.2 - Problem 28 is the chain rule for the derivative of...Ch. 14.2 - Using the definition of ez by its power series...Ch. 14.2 - Using the definitions of sin...Ch. 14.2 - Using series you know from Chapter 1, write the...Ch. 14.2 - Using series you know from Chapter 1, write the...Ch. 14.2 - Using series you know from Chapter 1, write the...Ch. 14.2 - Using series you know from Chapter 1, write the...Ch. 14.2 - Using series you know from Chapter 1, write the...Ch. 14.2 - Using series you know from Chapter 1, write the...Ch. 14.2 - Using series you know from Chapter 1, write the...Ch. 14.2 - Using series you know from Chapter 1, write the...Ch. 14.2 - Using series you know from Chapter 1, write the...Ch. 14.2 - In Chapter 12, equations (5.1) and (5.2), we...Ch. 14.2 - Prob. 44PCh. 14.2 - Prob. 45PCh. 14.2 - Prob. 46PCh. 14.2 - Prob. 47PCh. 14.2 - Using polar coordinates (Problem 46), find out...Ch. 14.2 - Prob. 49PCh. 14.2 - Using polar coordinates (Problem 46), find out...Ch. 14.2 - Prob. 51PCh. 14.2 - Prob. 52PCh. 14.2 - Using polar coordinates (Problem 46), find out...Ch. 14.2 - Show that the following functions are harmonic,...Ch. 14.2 - Show that the following functions are harmonic,...Ch. 14.2 - Show that the following functions are harmonic,...Ch. 14.2 - Show that the following functions are harmonic,...Ch. 14.2 - Show that the following functions are harmonic,...Ch. 14.2 - Show that the following functions are harmonic,...Ch. 14.2 - Show that the following functions are harmonic,...Ch. 14.2 - Show that the following functions are harmonic,...Ch. 14.2 - Show that the following functions are harmonic,...Ch. 14.2 - Show that the following functions are harmonic,...Ch. 14.2 - It can be shown that, if u(x,y) is a harmonic...Ch. 14.3 - Evaluate the following line integrals in the...Ch. 14.3 - Evaluate the following line integrals in the...Ch. 14.3 - Evaluate the following line integrals in the...Ch. 14.3 - Evaluate the following line integrals in the...Ch. 14.3 - Evaluate the following line integrals in the...Ch. 14.3 - Evaluate the following line integrals in the...Ch. 14.3 - Evaluate the following line integrals in the...Ch. 14.3 - Evaluate the following line integrals in the...Ch. 14.3 - Evaluate the following line integrals in the...Ch. 14.3 - Evaluate the following line integrals in the...Ch. 14.3 - Evaluate C(z3)dz where C is the indicated closed...Ch. 14.3 - 01+2iz2dz along the indicated paths:Ch. 14.3 - In Chapter 6, Section 11, we showed that a...Ch. 14.3 - In finding complex Fourier series in Chapter 7, we...Ch. 14.3 - If f(z) is analytic on and inside the circle z=1,...Ch. 14.3 - If f(z) is analytic in the disk z2, evaluate...Ch. 14.3 - Use Cauchys theorem or integral formula to...Ch. 14.3 - Use Cauchys theorem or integral formula to...Ch. 14.3 - Use Cauchys theorem or integral formula to...Ch. 14.3 - Use Cauchys theorem or integral formula to...Ch. 14.3 - Differentiate Cauchys formula (3.9) or (3.10) to...Ch. 14.3 - Use Problem 21 to evaluate the following...Ch. 14.3 - Use Problem 21 to evaluate the following...Ch. 14.3 - Use Problem 21 to evaluate the following...Ch. 14.4 - Show that the sum of a power series which...Ch. 14.4 - Show that equation ( 4.4 ) can be written as...Ch. 14.4 - For each of the following functions find the first...Ch. 14.4 - For each of the following functions find the first...Ch. 14.4 - For each of the following functions find the first...Ch. 14.4 - For each of the following functions find the first...Ch. 14.4 - For each of the following functions find the first...Ch. 14.4 - For each of the following functions find the first...Ch. 14.4 - For each of the following functions, say whether...Ch. 14.4 - For each of the following functions, say whether...Ch. 14.4 - For each of the following functions, say whether...Ch. 14.4 - For each of the following functions, say whether...Ch. 14.5 - If C is a circle of radius about z0, show that...Ch. 14.5 - Verify the formulas (4.3) for the coefficients in...Ch. 14.5 - Obtain Cauchys integral formula ( 3.9 ) from the...Ch. 14.6 - Find the Laurent series for the following...Ch. 14.6 - Find the Laurent series for the following...Ch. 14.6 - Find the Laurent series for the following...Ch. 14.6 - Find the Laurent series for the following...Ch. 14.6 - Find the Laurent series for the following...Ch. 14.6 - Find the Laurent series for the following...Ch. 14.6 - Find the Laurent series for the following...Ch. 14.6 - Find the Laurent series for the following...Ch. 14.6 - Find the Laurent series for the following...Ch. 14.6 - Show that rule B is correct by applying it to...Ch. 14.6 - Derive (6.2) by using the limit definition of the...Ch. 14.6 - Prove rule C for finding the residue at a multiple...Ch. 14.6 - Prove rule C by using (3.9). Hints: If f(z) has a...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Prob. 33PCh. 14.6 - Find the residues of the following functions at...Ch. 14.6 - Find the residues of the following functions at...Ch. 14.6 - For complex z,Jp(z) can be defined by the series...Ch. 14.6 - The gamma function (z) is analytic except for...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - The values of the following integrals are known...Ch. 14.7 - In Example 4 we stated a rule for evaluating a...Ch. 14.7 - Using the rule of Example 4 (also see problem 21),...Ch. 14.7 - Using the rule of Example 4 (also see problem 21),...Ch. 14.7 - Using the rule of Example 4 (also see problem 21),...Ch. 14.7 - Using the rule of Example 4 (also see problem 21),...Ch. 14.7 - Using the rule of Example 4 (also see problem 21),...Ch. 14.7 - Using the rule of Example 4 (also see problem 21),...Ch. 14.7 - Using the rule of Example 4 (also see problem 21),...Ch. 14.7 - Using the rule of Example 4 (also see problem 21),...Ch. 14.7 - (a) By the method of Example 2 evaluate 0dx1+x4....Ch. 14.7 - Use the method of Problem 30(c) to evaluate...Ch. 14.7 - Use the method of Problem 30(c) and the contour...Ch. 14.7 - Evaluate the following integrals by the method of...Ch. 14.7 - Evaluate the following integrals by the method of...Ch. 14.7 - Evaluate the following integrals by the method of...Ch. 14.7 - Evaluate the following integrals by the method of...Ch. 14.7 - (a) Show that epx1+exdx=sinp for 0p1. Hint: Find...Ch. 14.7 - Using the same contour and method as in Problem...Ch. 14.7 - Evaluate e2x/3coshxdx. Hint: Use a rectangle as in...Ch. 14.7 - Evaluate 0xdxsinhx. Hint: First find the to ...Ch. 14.7 - The Fresnel integrals, 0usinu2du and 0ucosu2du,...Ch. 14.7 - If F(z)=f(z)/f(z) (a) show that the residue of...Ch. 14.7 - By using theorem (7.8), show that z3+z2+9=0 has...Ch. 14.7 - The fundamental theorem of algebra says that every...Ch. 14.7 - As in Problem 43 find out in which quadrants the...Ch. 14.7 - As in Problem 43 find out in which quadrants the...Ch. 14.7 - As in Problem 43 find out in which quadrants the...Ch. 14.7 - As in Problem 43 find out in which quadrants the...Ch. 14.7 - As in Problem 43 find out in which quadrants the...Ch. 14.7 - As in Problem 43 find out in which quadrants the...Ch. 14.7 - Use (7.8) to evaluate...Ch. 14.7 - Use (7.8) to evaluate z3dz1+2z4 around z=1.Ch. 14.7 - Use (7.8) to evaluate z3+4zz4+8z2+16dz around the...Ch. 14.7 - Use (7.8) to evaluate Csec2(z/4)dz1tan(z/4), where...Ch. 14.7 - Find the inverse Laplace transform of the...Ch. 14.7 - Find the inverse Laplace transform of the...Ch. 14.7 - Find the inverse Laplace transform of the...Ch. 14.7 - Find the inverse Laplace transform of the...Ch. 14.7 - Find the inverse Laplace transform of the...Ch. 14.7 - Find the inverse Laplace transform of the...Ch. 14.7 - Find the inverse Laplace transform of the...Ch. 14.7 - Find the inverse Laplace transform of the...Ch. 14.7 - Find the inverse Laplace transform of the...Ch. 14.7 - Find the inverse Laplace transform of the...Ch. 14.7 - Find the inverse Laplace transform of the...Ch. 14.7 - In equation (7.18), let u(x) be an even function...Ch. 14.8 - Let f(z) be expanded in the Laurent series that is...Ch. 14.8 - (a) Show that if f(z) tends to a finite limit as z...Ch. 14.8 - Find out whether infinity is a regular point, an...Ch. 14.8 - Find out whether infinity is a regular point, an...Ch. 14.8 - Find out whether infinity is a regular point, an...Ch. 14.8 - Find out whether infinity is a regular point, an...Ch. 14.8 - Find out whether infinity is a regular point, an...Ch. 14.8 - Find out whether infinity is a regular point, an...Ch. 14.8 - Find out whether infinity is a regular point, an...Ch. 14.8 - Find out whether infinity is a regular point, an...Ch. 14.8 - Find out whether infinity is a regular point, an...Ch. 14.8 - Find out whether infinity is a regular point, an...Ch. 14.8 - Prob. 13PCh. 14.8 - Evaluate the following integrals by computing...Ch. 14.8 - Evaluate the following integrals by computing...Ch. 14.8 - Observe that in Problems 14 and 15 the sum of the...Ch. 14.9 - In these problems you should be able to make rough...Ch. 14.9 - For each of the following functions w=f(z)=u+iv,...Ch. 14.9 - For each of the following functions w=f(z)=u+iv,...Ch. 14.9 - For each of the following functions w=f(z)=u+iv,...Ch. 14.9 - For each of the following functions w=f(z)=u+iv,...Ch. 14.9 - For each of the following functions w=f(z)=u+iv,...Ch. 14.9 - For each of the following functions w=f(z)=u+iv,...Ch. 14.9 - For each of the following functions w=f(z)=u+iv,...Ch. 14.9 - Describe the Riemann surface for w=z3Ch. 14.9 - Describe the Riemann surface for w=zCh. 14.9 - Describe the Riemann surface for w=lnzCh. 14.9 - If w=f(z)=u(x,y)+iv(x,y),f(z) analytic, defines a...Ch. 14.9 - Verify the matrix equation dudv=Jdxdy, where J is...Ch. 14.9 - We have discussed the fact that a conformal...Ch. 14.9 - Compare the directional derivative...Ch. 14.10 - Prove the theorem stated just after (10.2) as...Ch. 14.10 - Assuming from electricity the equations...Ch. 14.10 - A fluid flow is called irrotational if V=0 where...Ch. 14.10 - Let a flat plate in the shape of a quarter-circle,...Ch. 14.10 - Consider a capacitor made of two very large...Ch. 14.10 - Prob. 6PCh. 14.10 - Use the mapping function w=z2 to find the...Ch. 14.10 - Prob. 8PCh. 14.10 - Find and sketch the streamlines for the flow of...Ch. 14.10 - Find and sketch the streamlines for the indicated...Ch. 14.10 - For w=ln[(z+1)/(z1)], show that the images of u=...Ch. 14.10 - Use the results of Problem 11 to solve the...Ch. 14.10 - Let the figure in Problem 12 represent (the cross...Ch. 14.10 - In the figure in Problem 12, let z=1 be a source...Ch. 14.10 - In Problem 14, the streamlines were the images of...Ch. 14.10 - Two long parallel cylinders form a capacitor. (Let...Ch. 14.11 - In Problems 1 and 2, verify that the given...Ch. 14.11 - In Problems 1 and 2, verify that the given...Ch. 14.11 - Liouvilles theorem: Suppose f(z) is analytic for...Ch. 14.11 - Use Liouvilles theorem (Problem 3 ) to prove the...Ch. 14.11 - In Problems 5 to 8, find the residues of the given...Ch. 14.11 - In Problems 5 to $8,$ find the residues of the...Ch. 14.11 - In Problems 5 to 8, find the residues of the given...Ch. 14.11 - In Problems 5 to $8,$ find the residues of the...Ch. 14.11 - In Problems 9 to 10, use Laurent series to find...Ch. 14.11 - In Problems 9 to $10,$ use Laurent series to find...Ch. 14.11 - Find the Laurent series of f(z)=ez/(1z) for z1 and...Ch. 14.11 - Let f(z) be the branch of z21 which is positive...Ch. 14.11 - In Problems 13 and $14,$ find the residues at the...Ch. 14.11 - In Problems 13 and 14, find the residues at the...Ch. 14.11 - In Problem 15 to 20, evaluate the integrals by...Ch. 14.11 - In Problem 15 to 20, evaluate the integrals by...Ch. 14.11 - In Problem 15 to 20, evaluate the integrals by...Ch. 14.11 - In Problem 15 to $20,$ evaluate the integrals by...Ch. 14.11 - In Problem 15 to 20, evaluate the integrals by...Ch. 14.11 - In Problem 15 to $20,$ evaluate the integrals by...Ch. 14.11 - Verify the formulas in Problem 21 to 27 by contour...Ch. 14.11 - Verify the formulas in Problem 21 to 27 by contour...Ch. 14.11 - Verify the formulas in Problem 21 to 27 by contour...Ch. 14.11 - Verify the formulas in Problem 21 to 27 by contour...Ch. 14.11 - Verify the formulas in Problem 21 to 27 by contour...Ch. 14.11 - Verify the formulas in Problem 21 to 27 by contour...Ch. 14.11 - Verify the formulas in Problem 21 to 27 by contour...Ch. 14.11 - Evaluate 0xlnxdx(1+x)2 by using the contour of...Ch. 14.11 - Evaluate 0(lnx)21+x2dx by using the contour of...Ch. 14.11 - Show that PV0cos(lnx)x2+1dx=2cosh(/2) by...Ch. 14.11 - As in Section 7, find out how many roots the...Ch. 14.11 - As in Section 7, find out how many roots the...Ch. 14.11 - As in Section 7, find out how many roots the...Ch. 14.11 - As in Section 7, find out how many roots the...Ch. 14.11 - Show that the Cauchy-Riemann equations [see (2.2)...Ch. 14.11 - Show that a harmonic function u(x,y) is equal at...Ch. 14.11 - A (nonconstant) harmonic function takes its...Ch. 14.11 - Show that a Dirichlet problem (see Chapter 13,...Ch. 14.11 - Use the following sequence of mappings to find the...Ch. 14.11 - Use L13 of the Laplace transform table to find the...Ch. 14.11 - Evaluate by contour integration 0cos2(/2)122d....
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- 6) A farmer has 60 acres on which to plant oats or corn. Each acre of oats requires 100 lbs of fertilizer and 1 hour of labor. Each acre of corn requires 50 lbs of fertilizer and 2 hours of labor. The farmer has 5000 lbs of fertilizer and 100 hours available for labor. If the profit is $60 from each acre of oats and $100 from each acre of corn, what planting combination will produce the greatest total profit? a) Fill in the following chart to help organize the information given in the problem: Oats Labor Fertilizer Land Profit b) Write down the question of interest. Corn Available c) Define variables to answer the question of interest. Call these x and y. d) Write the objective function to answer the question of interest. e) List any constraints given in the problem.arrow_forwardAn article appeared in the Journal of Gambling Issues, in which the authors looked at random samples of Ontario residents who (i) have not completed some form of post-secondary education and (ii) have completed some form of post-secondary education. A code of 0 indicates the person does not have a gambling problem, a code of 1 indicates the person does have a gambling problem. The data is found in the accompanying data file. Download.csv file To count the frequencies of 0 and 1 in each sample, use the table(your_dataset_name$ column's name) function. Make sure to replace "your_dataset_name" with the actual name of your data file and specify the correct column name. For example: table(file60c5d1286c735$ CompletedPSEducation) Let PNOPS represent the proportion of persons not completing some form of post-secondary education who have a gambling problem, and PPs be the proportion of persons having completed post-secondary education who have a gambling problem. (a) Find a 92% confidence…arrow_forwardWe consider a (European) call option on a stock with expiration in 3 months and strike price $10. The annual interest rate on the market is r = 4%. The current price of the stock is $10 and we assume that the stock follows a geometric Brownian motion (Black-Scholes) model with parameters = 6% and σ = 0.2. (a) Determine the price Fo of this option at time t = : 0 (today). (b) Using the formulas provided in the lecture videos, calculate the value of each of the Greeks for this option. Namely, calculate A, T, v, О, p. (c) Find a formula for the change of the option price with respect to a change in the af (St, t) Әк strike price. In other words, determine (d) For each of the suggested modifications below, use an approximation to determine the change in the price of the option above without actually recalculating the price. For each one, provide an intuitive argument to explain why the price increases or decreases. (i) The rate of return μ decreases to 5%. (ii) The interest rate r…arrow_forward
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