omplex Homework Summer 24 Based on Brown hurchill 7th Edition June 2, 24 ontents hw, omplex Arithmetic, onjugates, Polar Form 2 2 hw2 nth roots, Domains, Functions 2 3 hw3 Images, Transformations 3 4 hw4 Limits 3 5 hw5 Unbounded 4 6 hw6 Derivatives, auchy-riemann 4 7 hw7 Exp Log 5 8 hw8 Log log 5 9 hw9 Principal values, Integrals over a Real Variable 6 hw ontour Integrals 6 hw More on ontour Integrals 7 2 hw2 Path independence 7 3 hw3 auchy Goursat 8 4 hw4 Applications of auchy Integral Formula 8 5 hw5 Liouville 9 6 hw6 Series 9 7 hw7 Taylor Series 8 hw8 Laurent Series 9 hw9 Derivative of Series, Substituting, Poles, Residues 2 hw2 Singular points 2 2 hw2 Residues, Poles, Order of a Pole 2
22 hw22 omputing Integrals 3 23 hw23 Poles Zeros 3 24 hw24 ool Integrals 3 These are problems will be due both daily at the end of classes. This PDF file was created on June 2, 24. hw, omplex Arithmetic, onjugates, Polar Form. (B3.) Reduce each of these 3 expressions to a real number + 2i 3 4i + 2 i 5i 5i ( i)(2 i)(3 i) ( i) 4 2. (B4.) In each case locate + 2 2 vectorially = 2i, 2 = 2 3 i = ( 3, ), 2 = ( 3, ) = ( 3, ), 2 = (, 4) = x + iy, 2 = x iy 3. (B4.4) Sketch the set of points determined by each equation + i = + i 3 + 4i 4 4. (B5.3,4) Verify 2 = 2, 2 = 2, 2 3 = 2 3 4 = 4. 5. (B5.5) Verify 2 = 2 ( 2 ) 6. (B5.5) Show that the hyperbola x 2 y 2 = can be written 2 + 2 = 2 7. (B7.) Find the principal argument Arg for both = 8. (B7.2) Show e iθ = e iθ = e iθ i 2 2i = ( 3 i) 6 9. (B7.5) Use de Moivre s formula to derive the following trig identities. cos 3θ = cos 3 θ 3 cos θ sin 2 θ = 4 cos 3 θ 3 cos θ sin 3θ = 3 cos 2 θ sin θ sin 3 θ = 3 sin θ 4 sin 3 θ 2 hw2 nth roots, Domains, Functions. (B7.7) Show if R > R 2 > then Arg( 2 ) = Arg + Arg 2 2. (B9.) Find the square roots of 2i i 3 expressed in rectangular form 3. (B9.3) Find all of the roots in rectangle coordinates of ( ) /3 8 /6. 2
4. (B9.6) Find the 4 roots of p() = 4 + 4 = use them to factor p() into quadratic factors with real coefficients. 5. (B.-3) Sketch the 6 sets determine which are domains, which are bounded, which are neither open nor closed: 2 + i 2 + 3 > 4 I > I = arg π/4 ( ) 4 6. (B.4) Find the closure of the 4 sets: π < arg < π ( ) R < R( ) 2 R( 2 ) > 7. (B.) For each function, describe the domain that is understood: f() = 2 + f() = Arg( ) f() = + f() = 2 8. (B.2) Write 3 + + as u(x, y) + iv(x, y) 9. (B.3) Write simplify f() = x 2 y 2 2y + i(2x 2xy) in terms of using x = ( + )/2 y = ( )/2i. (B.4) Write f() = + / ( ) in the form u(r, θ) + iv(r, θ) 3 hw3 Images, Transformations. (B3.) Find a domain in the -plane whose image under the transformation w = 2 is the square domain in the w-plane bounded by the lines u =, u = 2, v =, v = 2 2. (B3.3) Sketch the region onto which the sector r, θ π/4 is mapped by the 3 transformations w = 2, w = 3, w = 4 3. (B3.4) Show that lines ay = x (a ) are mapped onto the spirals ρ = exp(aθ) under the transformation w = exp, where w = ρ exp(iφ) 4. (B3.7) Find the image of the semi-infinite strip x, y π under the transformation w = exp. Label the corresponding portions of the boundaries. 5. (B3.8) Graphically indicate the vector fields represented by w = i w = / 4 hw4 Limits. (B7.3) Find the limits. n is a positive integer, P () Q() are polynomials with Q( ) lim n ( ) i 3 lim i + i P () lim Q() 2. (B7.5) Show that the following limit does not exist lim ( )2 3
3. (B7.) Use a theorem to show: lim 4 2 ( ) 2 = 4 lim 4. (B7.) Suppose ad bc let: Use a theorem to show 2 + = lim ( ) 3 = T () = a + b c + d lim T () = (if c = ) lim T () = a (if c ) lim T () = (if c ) c d/c 5 hw5 Unbounded. (B7.3)( Show that a set S is unbounded if only if every neighborhood of the point at infinity contains at least one point of S. 6 hw6 Derivatives, auchy-riemann. (B9.) Find f () when f() = 3 2 2 + 4 f() = ( 4 2 ) 3 f() = 2 + ( 2 ) f() = ( + 2 ) 4 2 ( ) 2. (B9.2) Show if P () = a + a + a 2 2 + + a n n then P () = a + 2a 2 + + na n hence a = P (), a = P ()!, a 2 = P (),... a n = P (n) () 2! n! 3. (B9.9) Let f denote the function whose values are { f() = 2 / when when = Show that if =, then w/ = at each nonero point on the real imaginary axes in the or x y-plane. Then show then w/ = at each nonero point along the line y = x. onclude that f () does not exist. 4. (B22.6) Let f denote the function above. Show that the auchy-riemann equations are satisfied at the origin = (, ) 5. (B22.) Use a theorem to show that f () does not exist at any point for each function: f() = f() = f() = 2x + ixy 2 f(x) = e x e iy 6. (B22.2) Use a theorem to show that f () its derivative f () exist everywhere find f (). f() = i + 2 f() = e x e iy f() = 3 f() = cos x cosh y i sin x sinh y 4
7. Extra redit (B22.) Recall = x + iy implies x = ( + )/2 y = ( )/2i. Use the formal chain rule to show F = F x x + F y y = 2 ( F x + i F y ) Define the operator = 2 ( x + i y ) apply it to u(x, y)+iv(x, y) to obtain the complex form of the auchy-reimann equations f/ =. 7 hw7 Exp Log. (B28.) Show that exp(2 ± 3πi) = e 2, exp((2 + πi)/4) = ( + i) e/2 exp( + πi) = exp. 2. (B28.2) State why the function 2 2 3 e + e is entire. 3. (B28.3) Show f() = exp is not analytic anywhere. 4. (B28.7) Prove exp( 2) < if only if R >. 5. (B28.8) Find all values of such that e = 2, or e = + 3i or exp(2 ) = 6. (B28.) Show that if e is real, then I = nπ (n =, ±, ±2,... ). If e is pure imaginary, what restriction is placed on? 7. (B3.) Show that Log( ei) = π 2 i Log( i) = 2 ln 2 π 4 i. 8 hw8 Log log. (B3.2) Verify for n =, ±, ±2,...:. log e = + 2nπi log i = (2n + 2 )πi log( + 3i) = ln 2 + 2(n + 3 )πi 2. (B3.3) Show that Log( + i) 2 = 2 Log( + i) Log( + i) 2 2 Log( + i). 3. (B3.5) Show that the set of values of log(i /2 ) is {(n + 4 )πi : n =, ±, ±2,... } that the same is true of (/2) log i. 4. (B3.6) Given that the branch log = ln r + iθ (r >, α < θ < α + 2π) of the logarithmic function is analytic at each point in the stated domain, obtain its derivative by differentiating each side of the identity exp(log ) = using the chain rule. 5. (B3.7) Find all the roots of the equation log = iπ/2. 6. (B3.9) Show that Log( i) is analytic everywhere except on the half line y = (x ). Show Log( + 4) 2 + i is analytic everywhere except at the points ±( i)/ 2 on the portion x 4 of the real axis. 5
9 hw9 Principal values, Integrals over a Real Variable. (B3.) Show if R > R 2 > then Log( 2 ) = Log + Log 2. 2. (B3.2) Show that for any two complex numbers 2, Log( 2 ) = Log + Log 2 + 2Nπi where N has one of the values, ±. 3. (B32.) Show that when n =, ±, ±2... ( + i) i = exp( π 4 + 2nπ) exp( i 2 ln 2) ( )/π = e (2n+)i 4. (B32.2) Find the principal values of each expression: i i [ e 2 ( 3i)] 3πi ( i) 4i 5. (B32.5) Show that the principal n-th root of a nonero complex number is the same as the principal value of /n that was previously defined. 6. (B32.8) Let c, d, be complex numbers with. Prove that if all the powers involved are principal values, then c = c ( c ) n = cn (n =, 2,... ) c d = c+d 7. (B37.2) Evaluate c d = c d. 2 ( t i)2 dt π/6 e i2t dt e t dt (R > ) 8. (B37.5) Let w(t) be a continuous complex-valued funtion of t defined on an interval a t b. By considering the special case w(t) = e it on the interval t 2π, show that it is not always true that there is a number c in the interval a < t < b such that hw ontour Integrals b a w(t) dt = w(c)(b a). (B38.2) Let denote the right-h half of the circle = 2, in the counterclockwise direction note that two parametric representations for are Verify that Z(y) = [φ(y)], where = (θ) = 2e iθ ( π 2 θ π 2 ) = Z(y) = 4 y 2 + iy ( 2 y 2) φ(y) = arctan y 4 y 2 ( π 2 arctan t π 2 ) Also, show that this function φ has a positive derivative, as required in the conditions following (9) Sec 38. 6
2. (B4.,2,3,5,6) Evaluate for the given f() contour f() = ( + 2)/ is = 2e iθ ( θ π) f() = ( + 2)/ is = 2e iθ (π θ 2π) f() = ( + 2)/ is = 2e iθ ( θ 2π) f() = + is = + e iθ (π θ 2π) f() = + is = t ( t 2) f() = π exp(π) is square from,, + i, i f() d f() = is arbitrary curve from to 2 f() = +i is = positively oriented use branch exp[( + i) log ] ( >, < arg < 2π) 3. (B4.) Let denote the circle = R taken counterclockwise. Use the parametric representation = + Re iθ ( π θ π) for to derive the following integration formula s: d = 2πi ( ) n d = (n = ±, ±2,... ) hw More on ontour Integrals. (B4.4) Let R denote the upper half of the circle = R (R > 2), taken in the counterclockwise direction. Show that 2 2 R 4 + 5 2 + 4 d πr(2r2 + ) (R 2 )(R 2 4) 2. (B43.) Use an antiderivative to show that, for every contour extending from a point to a point 2, n d = n + (n+ 2 n+ ) (n =,,... ) 3. (B43.2) By finding an antiderivative, evaluate each of these integrals, where the path is any contour between the indicated limits of integration. i/2 i e π d π+2i cos( 3 2 ) d ( 2) 3 d 2 hw2 Path independence. (B43.3) Use a theorem to show ( ) n d = (n = ±, ±2,... ) when is any closed contour which does not pass through the point. 2. (B43.4) Let, (resp. 2 ), be any contour from = 3 to = 3 that except for its end points, lies above (resp. below) the x-axis. Find an antiderivative F 2 () of the branch f 2 () of /2 = re iθ/2 (r >, π 2 < θ < 5π 2 ) 7
to show that the integral 2 /2 d has value 2 3( + i). Note that the value of the integral of the function /2 = re iθ/2 around the closed contour 2 in that example is, therefore 4 3 given that /2 d = 2 3( + i). (Lots of parts from example 43.4.) 3 hw3 auchy Goursat. (B46.) Apply the auchy-goursat theorem to show that f() d = when the contour is the circle =, in either direction when f() = 2 3 f() = e f() = 2 + 2 + 2 f() = sech f() = tan f() = Log( + 2) 2. (B46.2) Let be the positively oriented circle = 4 let 2 be the positively oriented boundary of the square whose sides lie along the lines x = ±, y = ±. Point out why f() d = f() d 2 when f() = 3 2 + f() = + 2 sin(/2) f() = e 3. (B46.3) If is the boundary of the rectangle x 3, y 2, described in the positive sense, then ( 2 i) n = 2πi when n = when n = ±, ±2,... 4. (B46.4) Extra redit???? 4 hw4 Applications of auchy Integral Formula. (B48.abc) Let denote the positively oriented boundary of the square whose sides lie along the lines x = ±2, y = ±2. Evaluate the integrals e d (πi/2) cos d ( 2 + 8) d 2 + 8
2. (B48.2) Find the integral of g() around the circle i = 2 in the positive sense when g() = /( 2 + 4) when g() = /( 2 + 4) 2. 3. (B48.3) Let be the circle = 3 decribed in the positive sense. Show that if 2 2 2 g(w) = d ( w 3) w then g(2) = 8πi. What is the value of g(w) when w > 3? 4. (B48.7) Let be the unit circle = e iθ ( π θ π). First show that for any real constant a, e a d = 2πi Then write this integral in terms of θ to derive the integration formula π e a cos θ cos(a sin θ) dθ = π 5. (B48.6) Extra redit???? Let f denote a function that is continuous on a simple closed contour. Prove the function g() = f(ξ) dξ 2πi ξ is analytic as each point interior to that g () = 2πi at such a point. 5 hw5 Liouville f(ξ) dξ (ξ ) 2. (B5.) Let f be an entire function such that f() A for all, where A is a fixed positive number. Show that f() = a, where a is a complex constant. [Hint: use auchy s inequality to show f () is ero.] 2. (B5.) Suppose f() is entire that the harmonic function u(x, y) = Rf() has an upper bound u : that is, u(x, y) u for all points (x, y) in the xy-plane. Show that u(x, y) must be constant throughout the plane. [Hint: use Liouville s theorem on exp(f()).] 3. (B5.4,5) Let a function f be continuous in a closed bounded region R, let it be analytic not constant throughout the interior of R. Assuming f() anywhere in R, prove that f() has a minimum value m in R which occurs on the boundary of R never in the interior. [Hint: look at /f().] Use the function f() = to show that the condition f() anywhere is necessary for this conclusion. 6 hw6 Series. (B52.6) Show if n= n = S, then n= n = S. 2. (B52.7) Show for any complex number c Show if n= n = S, then n= c n = cs. 3. (B52.8) Show if n= n = S n= w n = T, then n= ( n + w n ) = S + T. 9
7 hw7 Taylor Series. (B54.2) Obtain the Taylor e = e ( ) n two ways. First using f (n) () second by using e = ee. 2. (B54.3) Find the Maclaurin series expansion for the function f() = n! ( < ) 4 + 9 = 9 + 4 /9 3. (B54.5) Derive the Maclaurin series for cos by showing f (2n) () = ( ) n f (2n+) () = by using cos = (e i + e i )/2. 4. (B54.) Show when, 5. (B54.3) Show that when < < 4, 8 hw8 Laurent Series e 2 = 2 + + 2! + 3! + 2 4! + sin( 2 ) 4 = 2 2 3! + 6 5! + 7! 4 2 = 4 + n 4 n+2. (B56.) Find the Laurent series that represents the function f() = 2 sin(/ 2 ) in the domain < <. 2. (B56.2) Derive the Laurent series representation [ e ( + ) 2 = ] ( + ) n e (n + 2)! + + + ( + ) 2 3. (B56.3) Find a representation for the function f() = + = + (/) in negative powers of that is valid for < <. 4. (B56.4) Give two Laurent series expansions in powers of for the function f() = /[ 2 ( )] specify the regions in which the expansions are valid. [Hint: about ] 5. (B56.5) Represent the function f() = + by both its Maclaurin series (stating where it is valid) by a Laurent series in the domain < < 6. (B56.6) Show that when < < 2, ( )( 3) = 3 ( ) n 2 n+2 2( )
9 hw9 Derivative of Series, Substituting, Poles, Residues. (B6.) By differentiating the Maclaurin series representation obtain the expressions = n ( < ) ( ) 2 = (n + ) n ( < ) 2 ( ) 3 = (n + )(n + 2) n ( < ) 2. (B6.2) By substituting /( ) for in the expansion ( ) 2 = (n + ) n ( < ) found above, derive the Laurent series representation 2 = n=2 3. (B6.3) Find the Taylor series for the function ( ) n (n ) ( ) n ( < < ) = 2 + ( 2) = 2 + ( 2)/2 about the point = 2. Then by differentiating that series term by term, show that 2 = ( ) n (n + )( 2 4 2 )n ( 2 < 2) 4. (B6.) Use multiplication of series to show that e ( 2 + ) = + 2 5 6 2 + ( < < ) 5. (B6.3) Use division to obtain the Laurent series representation e = 2 + 2 72 3 + ( < < 2π) 6. (B64.) Find the residue at = of the functions + 2 cos( ) sin cot 4 sinh 4 ( 2 ) 7. (B64.2) Use auchy s residue theorem to evaluate the integral of each of these functions around the circle = 3 in the positive sense: exp( ) 2 exp( ) ( ) 2 2 exp( ) + 2 2 8. (B64.3) Use a theorem involving a single residue to evaluate the integral of each of these functions around the circle = 2 in the positive sense. 5 3 + 2
2 hw2 Singular points. (B65.) In each case, write the principal part of the function at its isolated singular point determine whether that point is a pole, a removable singular point or an essential singular pont. exp( ) 2 + sin cos (2 ) 3 2. (B65.2) Show that the singular point of each of the following functions is a pole. Determine the order m of the pole the corresponding residue B. cosh 3 exp(2) 4 exp(2) ( ) 2 3. (B65.3) Suppose f is analytic at write g() = f()/( ). Show that: (a) If f( ), then is a simple pole of g, with residue f( ). (b) Iff( ) =, then is a removable singular point of g. 2 hw2 Residues, Poles, Order of a Pole. (B65.4) Write the function as f() = f() = 8a3 2 ( 2 + a 2 ) 3 (a > ) φ() ( ai) 3 where φ() = 8a3 2 ( + ai) 3 Point out why φ() has a Taylor series representation about = ai, then use it to show that the principal part of f at that point is φ (ai)/2 ai + φ (ai) ( ai) 2 + φ(ai) ( ai) 3 = i/2 ai a/2 ( ai) 2 a 2 i ( ai) 3 2. (B67.) In each case, show that any singular point of the function is a pole. Determine the order m of the pole find the corresponding residue B 2 + 2 ( 2 + )3 exp 2 + π 2 3. (B67.2) Show that Res = Res =i /4 + = + i 2 ( >, < arg < 2π) Res =i /2 ( 2 + ) 2 = i 8 2 4. (B67.3) Find the value of the integral Log ( 2 + ) 2 = π + 2i 8 ( >, < arg < 2π) 3 3 + 2 ( )( 2 + 9) d taken counterclockwise around both circles 2 = 2 = 4 2
22 hw22 omputing Integrals. (B67.4) Find the value of the integral d 3 ( + 4) taken counterclockwise around both circles = 2 + 2 = 3 2. (B69.) Show that the point = is a simple pole of the function f() = csc = / sin by a theorem by computing the Laurent series. 3. (B69.3a) Show that Res ( sec ) = ( ) n+ n, where n = π + nπ (n =, ±, ±2,... = n 2 4. (B69.4a) Let denote the positively oriented circle = 2 evaluate the integral tan d 5. (B69.5) Let N denote the positive oriented boundary of the square whose edges lie along the lines where N is a positive integer. Show that x = ±(N + 2 )π y = ±(N + 2 )π N [ d 2 sin = 2πi N 6 + 2 n= ] ( ) n n 2 π 2 then using the fact that the value of this integral tends to ero as N tends to infinity, point out how it follows that ( ) n n 2 = π2 2 23 hw23 Poles Zeros n=. (B69.9) Let p q denote functions that are analytic at a point where p( ) q( ) =. Show that if the quotient p()/q() has a pole of order m at, then is a ero of order m of q. 24 hw24 ool Integrals. (B72.,2,4) Use residues to evaluate the following integrals dx x 2 + dx (x 2 + ) 2 x 2 dx (x 2 + )(x 2 + 4) 2. (B74.,2) Use residues to evaluate the following integrals cos x dx (x 2 + a 2 )(x 2 + b 2 ) (a > b > ) cos ax dx x 2 + (a > ) 3