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Solutions for Complex Variables and Applications | 9th Edition | ISBN: 9780073383170 | Authors: James Ward Brown

ISBN9780073383170

Solutions for Chapter 3: Elementary Functions

Solutions for Chapter 3

3.3.1) Show that . ) "' ., . , (a exp(~ .uri) = -e-: ( 2 +;ri) (b) exp 4 (c) exp(: +;ri) =-exp:.

3.3.15) If::: = - I - ./J;. then r = 2 and (;..) = -2n /3. Hence log(- I - fin =In 2 + i (- 2 ; + 211n) =In 2 + 2 (11 - ~) ni...

3.3.20) Show that ;r (al Log(-t'il =I - -::;-i:U>l Log( I - i l = - In 2 - -i.

3.3.32) Show that for any two nonzero complex numbers : 1 and ::2 Log(.:1 :2) = Log.: 1 + Log :2 + 2N ;r i where N has one of...

3.3.37) Show that . ( ;r ) ( In 2) (a) (I +i)' =exp - 4 +211;-r exp iT (11 = 0. 1. 2 .... ): I (/>) .. 2 ,. = cxpl(411 + I J;...

3.3.46) Give details in the derivation of expressions (2). Sec. 37. for the derivatives of sin:: and cos::.

3.3.62) Verify that the derivatives of sinh:: and cosh:: are as stated in equations (2). Sec. 39.

3.3.79) find all the values of ( /) tan- I (2j ): (c) cosh- 1 (-1): (d) tanh- 1 0. ( I) i A11.\. (Cl) /1 + 2' ;r + 21n:1(11=0...

3.3.2) State why the function f (::) = 1.c 2 - 3 - :e: + e< is entir

3.3.16) From expression (2) in Sec. 31. we find that log I= In I+ i(O + 211n) = 211ni As anticipated. Log I = 0. (II= 0. J. 2...

3.3.21) Show that I ;r U>l Log( I - i l = - In 2 - -i. - 2 4 (a) loge= I+ 211;ri (11 = 0. 1. +2 .... ): (bl logi = (111 + ~) ...

3.3.33) Verify expression (4). Sec. 34. for log(: 1 /: 2 l by (a) using the fact that arg(.: 1/.:2 ) = arg .: 1 - arg.:2 (Sec...

3.3.38) find the pri ocipal v;.ilue of (ii) (-i)i: (/J) [i<-1 - J3i) r, .. : (c) (I - i).i1 . Ans. (il)exp(;r/2): (/>)-exp(:!...

3.3.47) (a) With the aid of expression (4 ). Sec. 37. show that e'> ~: =cos :: 1 cos :: 2 - sin:: 1 sin :: 2 + i (sin:: 1 cos...

3.3.63) Prove that sin h 2:: = 2 si nh :: co sh :. by starting with (a) deli nitions (I). Sec. 39. of si nh :: and co sh::: (...

3.3.80) Solve the equation sin: = 2 for: by (Cl) equating real parts and then imaginary parts in that equation: (b) using exp...

3.3.3) Use the Cauchy-Riemann equations and the theorem in Sec. 21 to show that the function f (::) =exp; is not analytic an...

3.3.17) Observe that log(- I)= In J + i(n + 211n) = (211 + J )ni and that Log ( - 1) = 7r i. (11 = 0. I. 2 .... ) Special car...

3.3.22) Show that Log(i1 ) f- 3 Log i.

3.3.34) By choosing specitic nonzero values of : 1 and :2. show that expression (4). Sec. 34. for log(.: 1/:2 ) is not always...

3.3.39) de ti' . . I s 3 - . ( I.... I I ;;;; . ) ' , ' ' ., J;;- vSC nltlOn ( ). , CC. ). 0 :'. tO S1K)W t lat ( - + v .H .....

3.3.48) According to the flnal result in Exercise 2(/J). sin(::+ :: 2 ) = sin::cos:: 2 + cos::sin ::.:..By differentiating ea...

3.3.64) Show how identities (6) and (8) in Sec. 39 follow from identities (9) and ( 6). respectively. in Sec. 37.

3.3.81) Solve the equation cos: = ./2 for:.

3.3.4) Show in two ways that the function f(:) = exp(: 2 ) is entire. \Vhat is its derivative'.' A11s. /(::) = 2:: exp(: 2 ).

3.3.18) The identity ( J ) Log[( J + i ) 2 ] = 2 Log( J + i) is valid since Log(( I + i) 2 ] = Log C2i > = In 2 + i ~ and ( J...

3.3.23) Show that log (i 2 ) # 2 log i when the branch ( .3;r 11 ;r)

3.3.35) Show that property (6). Sec. 34. also holds when /1 is a negative integer. Do this by writing : 1,.,, = (:: l:'m ,--1...

3.3.40) Show that the result in Exercise 3 could have been obt;.iined by writing (a) (-1 + v 1Ji)>.: 2 =I (-1 + /3i) 1 2 1' a...

3.3.49) Verify identity (9) in Sec. 37 using (a) identity (6) and relations (3) in that section: ( b) the lemma in Sec. 28 an...

3.3.65) Write sinh:. = sinh(x + iy) and cosh:. = cosh(x + iy ). and then show how expressions (9) and (I 0) in Sec. 39 follow...

3.3.82) Derive expression (5). Sec. 40. for the derivative of sin- 1 :..

3.3.5) \Vrite lexp(2:: + i)I and lcxp(i:::.!)I in terms of x and y. Then show that . . ., .,, - .,\\ Jcxp(2::: + 1) + cxp(l:...

3.3.19) It is shown in Exercise 5. Sec. 33. chat I . I ,') I 1 ' 00(1 -) = - Oil/ e -, e in the sense that che sec of values ...

3.3.24) (a) Show that the two square roots of i arc Then show thal i :r /I e and i5.7:'t e log(ei.1'/I) = (211 + ~) ;ri (II =...

3.3.36) Lee: denote any nonzero complex number. written.: = n/H (-;r < (o-) ~ ;r ). and let /1 denote any tixed positive inte...

3.3.41) Sh<rn: that the pri11cipal 111h root of a nonzero complex number : 0 that was defined inSec. 10 is the same as the pr...

3.3.50) L:se identity (9) in Sec. 37 to show that (a) I+ tan2 .:: = sec2 .::: (b) I + rnt2 .:: = csc2 :.

3.3.66) Derive expression ( I I ) in Sec. 39 for lsi nh :: 1 2.

3.3.83) Derive expression (4). Sec. 40. for tan- 1 :..

3.3.6) Show that jexp(::: 2 ) I ::: exp( I::: I.!).

3.3.25) Given that the branch log:= lnr + iO(r > O.a < 0 0. a < arg: < a ~ 2;r) and using the chain rule.

3.3.42) Show that if: :f- 0 and a is a real number. then I:'' I = exp(il In I:. I) = 1:.1". where the principal value of I:. ...

3.3.51) Establish differentiation formulas (3) and (4) in Sec. 38.

3.3.67) Show that jsinh.\I.::: jcosh::I.::: coshx by using (a) identity (12). Sec. 39: (/J) the inequalities lsi nh .rl ::,: ...

3.3.84) Derive expression (7). Sec. 40. for the derivative of tan- 1 :..

3.3.7) Prove that jcxp(-2:::)10.

3.3.26) Show that a branch (Sec. 33) log ::: = In r + iO ( r > 0. a < 0 < u + 27 ) of the logarithmic function can be written...

3.3.43) Let c =ii +hi be a fixed complex number. where c :f= 0. I. 1 ..... and note th;.it i' is multiple-valued. \Vhat addit...

3.3.52) In Sec. 37. use expressions ( 13) and ( 14) to derive expressions ( 15) and ( 16) for I sin.:: 1 2 and jcos.::12. S11...

3.3.68) Show that (a) sinh(:: +:ri) = -sinh:::(/>) cosh(:: +;ri) - cosh::: (c) tanh<:: + ;ri) = tanh ::.

3.3.85) Derive expression (9). Sec. 40. for cosh- 1 :.

3.3.8) Find al I values of::: such that (a) e~ = -2: ( h) e~ = I + i: (c) cxp(2: - I) = I. A11.\. (a) ::: = ln2 + (211 + l);...

3.3.27) Find al I roots of the equation log:::. = i ;r /2. A11.\'. :.: = i.

3.3.44) Let c. c 1 c2 . and : denote complex numbers. where: -=j. 0. Prove th;.it if all of the powers involved are principal...

3.3.53) Point out how it follows from expressions (15) and ( 16) in Sec. 37 for I sin.:: 1 2 and jcos: 1 2 that (Cl) lsin.::I...

3.3.69) Give detai Is showing that the zeros of sinh:: and cosh:: arc as in the theorem in Sec. 39.

3.3.9) Show that exp(i :::) = exp(i;) if and only if::: = 11;r (11 = 0. I. 2 .... ). (Compare with Exercise 4. Sec. 29.)

3.3.28) ppose that the point:= x + iy lies in the horizontal strip a < y < a+ 2;r. Show that \Vhcn the branch log:.: = In r +...

3.3.45) Assuming that f' (:) exists. state the formula for the derivative of c / i.~ 1

3.3.54) th the aid of expressions ( 15) and ( 16) in Sec. 37 for lsin:l 2 and lcos.::12 show that (a) I sinh .rl ::: lsi n.::...

3.3.70) llsing the results proved in Exacise 8. locate all zeros and singularities of the hyperbolic tangent function.

3.3.10) (ii) Show that if e~ is real. then Im:: = 11;r (11 = 0. I. 2 .... ). (h) If e is pure imaginary. what restriction is ...

3.3.29) Show that (a) the function fC:.) = Log(::.. - i) is analytic everywhere except on the portion x ::: 0 of the liner= I...

3.3.55) (a) L'se delinitions (I). Sec. 37. of sin: and cos: to show that (/>) With the aid of the identity obtained in part (...

3.3.71) Show that tanh:: = -i tan(i .:- ). Sugge.\t ion: l; se idcnt itics (-t) in Sec. 39.

3.3.11) Describe the behavior of e: = eei 1 as (a) x tends to -ex:: (h) y tends to x.

3.3.30) Show in two ways that the function ln(x2 + y 2 ) is harmonic in every domain that docs not contain the origin.

3.3.56) L: se the Cauchy-Riemann equations and the theorem in Sec. 21 lo show that neither sin: nor cos: is an analytic funct...

3.3.72) Derive differentiation formulas ( 17 ). Sec. 39.

3.3.12) Write Re(t! 1 ':) in terms of x and y. Why is this function harmonic in every domain that does not contain the origin'!

3.3.31) Show that I , , Rellog(::..- ll]= -lnl

3.3.57) Use the rellection principle (Sec. 29) to show that for all:. (cl) sin.::= sin:: (/J) cos: =cos:.

3.3.73) .:se the reflection principle (Sec. 29) to show that for all::. (a) sinh:: = sinh :": (/J) cosh:: = cosh :.

3.3.13) Let the function/(:::)= 11(.r. y) +iv(.\", y) be analytic in some domain D. Stale why the functions l/(x. y) = e'111 ...

3.3.58) \Vith the aid of expressions (13) and (14) in Sec. 37. give direct verifications of the relations obtained in Exercis...

3.3.74) t;se the results in Exercise 12 to show that tanh:: = tanh: at points where cosh:: # 0.

3.3.14) Establish the identity (II = 0. I. 2 .... ) in the following way. (ll) t: se mathematical induction to show that it i...

3.3.59) Show that (a) cos(i.::) = cos(i ;) (/>) si n(i.::) = sin(i ;) for all.::: if and only if .:: = mri (11=0. 1. +2 .... ).

3.3.75) By accepting that the stated identity is valid when:.: is replaced by the real variable .r and using the lemma in Sec...

3.3.60) find all roots of the equation sin.:: = cosh4 by equating the real parts and then the imaginary parts of sin.:: and c...

3.3.76) Why is the function sinh(t'~) entire'! Write its real component as a function of x and y. and state why that function...

3.3.61) With the aid of expression ( 14). Sec. 37. show that the roots of the equation cos :: = 2 arc :: = 111;r + i cosh- 1 ...

3.3.77) By using one of the identities (9)and (IO) in Sec. 39 and then proceeding as in Exercise 15. Sec. 38. Jlnd all roots ...

3.3.78) find all roots of the equation cosh:.: = -2. (Compare this exercise with Exercise 16. Sec. 38.) An.\.:.: = ln(2 + J3J...

Complex Variables and Applications was written by and is associated to the ISBN: 9780073383170. Since 85 problems in chapter 3: Elementary Functions have been answered, more than 43368 students have viewed full step-by-step solutions from this chapter. Chapter 3: Elementary Functions includes 85 full step-by-step solutions. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Complex Variables and Applications, edition: 9.

Key Math Terms and definitions covered in this textbook
  • Cyclic shift

    S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

  • Distributive Law

    A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

  • Gauss-Jordan method.

    Invert A by row operations on [A I] to reach [I A-I].

  • Hermitian matrix A H = AT = A.

    Complex analog a j i = aU of a symmetric matrix.

  • Hilbert matrix hilb(n).

    Entries HU = 1/(i + j -1) = Jd X i- 1 xj-1dx. Positive definite but extremely small Amin and large condition number: H is ill-conditioned.

  • Hypercube matrix pl.

    Row n + 1 counts corners, edges, faces, ... of a cube in Rn.

  • Incidence matrix of a directed graph.

    The m by n edge-node incidence matrix has a row for each edge (node i to node j), with entries -1 and 1 in columns i and j .

  • Independent vectors VI, .. " vk.

    No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

  • Krylov subspace Kj(A, b).

    The subspace spanned by b, Ab, ... , Aj-Ib. Numerical methods approximate A -I b by x j with residual b - Ax j in this subspace. A good basis for K j requires only multiplication by A at each step.

  • Markov matrix M.

    All mij > 0 and each column sum is 1. Largest eigenvalue A = 1. If mij > 0, the columns of Mk approach the steady state eigenvector M s = s > O.

  • Multiplicities AM and G M.

    The algebraic multiplicity A M of A is the number of times A appears as a root of det(A - AI) = O. The geometric multiplicity GM is the number of independent eigenvectors for A (= dimension of the eigenspace).

  • Norm

    IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

  • Normal equation AT Ax = ATb.

    Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b - Ax) = o.

  • Particular solution x p.

    Any solution to Ax = b; often x p has free variables = o.

  • Positive definite matrix A.

    Symmetric matrix with positive eigenvalues and positive pivots. Definition: x T Ax > 0 unless x = O. Then A = LDLT with diag(D» O.

  • Projection matrix P onto subspace S.

    Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.

  • Rank r (A)

    = number of pivots = dimension of column space = dimension of row space.

  • Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.

    Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).

  • Spectral Theorem A = QAQT.

    Real symmetric A has real A'S and orthonormal q's.

  • Sum V + W of subs paces.

    Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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