 Chapter 13.13.5: Using the CauchyRiemann equations, show that eZ is entire.
 Chapter 13.13.1: (powersofi)Showthari2 = I, i3 = i, i4 = I, ;5 = i .... and IIi = ...
 Chapter 13.13.6: Prove that cos z, sin z, cosh z, sinh Z are entire functions.
 Chapter 13.13.2: Do these problems very carefully since polar forms will be needed f...
 Chapter 13.13.7: Principal Value Ln z. Find Ln z when z equals:
 Chapter 13.13.3: Find and sketch or graph the sets in the complex plane given by Iz ...
 Chapter 13.1: Add. subtract. multiply. and divide 26  7i and 3 + 4i as well as t...
 Chapter 13.13.4: Are the following functions analytic? [Use (1) or (7).]f(:;.) = :;.4
 Chapter 13.13.5: Values of eZ Compute eZ in the form u + iv and lezl, where ~ equals...
 Chapter 13.13.1: (Rotation) \1ultiplication by i is geometrically a counterclockwise...
 Chapter 13.13.6: Verify by differentiation that Re cos z and 1m sin z are harmonic.
 Chapter 13.13.2: Do these problems very carefully since polar forms will be needed f...
 Chapter 13.13.7: Principal Value Ln z. Find Ln z when z equals:
 Chapter 13.13.3: Find and sketch or graph the sets in the complex plane given by 1 ~...
 Chapter 13.2: Write the two given numbers in Prob. I in polar form. Find the prin...
 Chapter 13.13.4: Are the following functions analytic? [Use (1) or (7).] f(::.) = 1m...
 Chapter 13.13.5: Values of eZ Compute eZ in the form u + iv and lezl, where ~ equals...
 Chapter 13.13.1: (Dhision) Verify the calculation in (7).
 Chapter 13.13.6: Show that cosh z = cosh x cos Y + i sinh x sin ysinh z = sinh x cos...
 Chapter 13.13.2: Do these problems very carefully since polar forms will be needed f...
 Chapter 13.13.7: Principal Value Ln z. Find Ln z when z equals:
 Chapter 13.13.3: Find and sketch or graph the sets in the complex plane given by 0 <...
 Chapter 13.3: What is the triangle inequality? Its geometric meaning? Its signifi...
 Chapter 13.13.4: Are the following functions analytic? [Use (1) or (7).] e2 x(cos y ...
 Chapter 13.13.5: Values of eZ Compute eZ in the form u + iv and lezl, where ~ equals...
 Chapter 13.13.1: (Multiplication) If the product of two complex numbers is zero, sho...
 Chapter 13.13.6: Show that cosh (ZI + Z2) = cosh ZI cosh Z2 + sinh ZI sinh Z2 sinh (...
 Chapter 13.13.2: Do these problems very carefully since polar forms will be needed f...
 Chapter 13.13.7: Principal Value Ln z. Find Ln z when z equals:
 Chapter 13.13.3: Find and sketch or graph the sets in the complex plane given by7r<...
 Chapter 13.4: If you know the values of {,fl, how do you get from them the values...
 Chapter 13.13.4: Are the following functions analytic? [Use (1) or (7).]f(:;.) = I/O...
 Chapter 13.13.5: Values of eZ Compute eZ in the form u + iv and lezl, where ~ equals...
 Chapter 13.13.1: Show that: = x + iy is pure imaginary if and onJy if;: = :.
 Chapter 13.13.6: Show that cosh2 Z  sinh2 z. = 1
 Chapter 13.13.2: Do these problems very carefully since polar forms will be needed f...
 Chapter 13.13.7: Principal Value Ln z. Find Ln z when z equals:
 Chapter 13.13.3: Find and sketch or graph the sets in the complex plane given by 1m ...
 Chapter 13.5: State the definition of the derivative from memory. It looks simila...
 Chapter 13.13.4: Are the following functions analytic? [Use (1) or (7).] eX(cos y ...
 Chapter 13.13.5: Values of eZ Compute eZ in the form u + iv and lezl, where ~ equals...
 Chapter 13.13.1: (Laws for conjugates) Verify (9) for Zl = 24 + 10i.':2 = 4 + 6i.
 Chapter 13.13.6: Show that cosh2 Z + sinh2 Z = cosh 2z
 Chapter 13.13.2: Do these problems very carefully since polar forms will be needed f...
 Chapter 13.13.7: Principal Value Ln z. Find Ln z when z equals:
 Chapter 13.13.3: Find and sketch or graph the sets in the complex plane given by
 Chapter 13.6: What is an analytic function? How would you test for anal yticity?
 Chapter 13.13.4: Are the following functions analytic? [Use (1) or (7).]fez) = Arg 7TZ.
 Chapter 13.13.5: Values of eZ Compute eZ in the form u + iv and lezl, where ~ equals...
 Chapter 13.13.1: Let':l = 2 + 3i and Z2 = 4  5i. Showing the details of your work. ...
 Chapter 13.13.6: Function Values. Compute (in the form u + iv)
 Chapter 13.13.2: Do these problems very carefully since polar forms will be needed f...
 Chapter 13.13.7: Principal Value Ln z. Find Ln z when z equals:
 Chapter 13.13.3: Find and sketch or graph the sets in the complex plane given by
 Chapter 13.7: Can a function be differentiable at a pomt without being analytic t...
 Chapter 13.13.4: Are the following functions analytic? [Use (1) or (7).]
 Chapter 13.13.5: Values of eZ Compute eZ in the form u + iv and lezl, where ~ equals...
 Chapter 13.13.1: Let':l = 2 + 3i and Z2 = 4  5i. Showing the details of your work. ...
 Chapter 13.13.6: Function Values. Compute (in the form u + iv)
 Chapter 13.13.2: Do these problems very carefully since polar forms will be needed f...
 Chapter 13.13.7: Principal Value Ln z. Find Ln z when z equals:
 Chapter 13.13.3: Find and sketch or graph the sets in the complex plane given by
 Chapter 13.8: Are 1::1, .:. Re;::, 1m:: analytic? Give reason.
 Chapter 13.13.4: Are the following functions analytic? [Use (1) or (7).]
 Chapter 13.13.5: Real and Imaginary Parts. Find Re and 1m of e2z
 Chapter 13.13.1: Let':l = 2 + 3i and Z2 = 4  5i. Showing the details of your work. ...
 Chapter 13.13.6: Function Values. Compute (in the form u + iv)
 Chapter 13.13.2: Determine the principal value of the argument. I  i
 Chapter 13.13.7: Principal Value Ln z. Find Ln z when z equals:
 Chapter 13.13.3: Find and sketch or graph the sets in the complex plane given by
 Chapter 13.9: State the definitions of eZ , cos z. sin ;::. cosh z. sinh;:: and t...
 Chapter 13.13.4: Are the following functions analytic? [Use (1) or (7).]
 Chapter 13.13.5: Real and Imaginary Parts. Find Re and 1m of ez3
 Chapter 13.13.1: Let':l = 2 + 3i and Z2 = 4  5i. Showing the details of your work. ...
 Chapter 13.13.6: Function Values. Compute (in the form u + iv)
 Chapter 13.13.2: Determine the principal value of the argument. 20 + ;,  20  ;
 Chapter 13.13.7: ll Values of In z. Find all values and graph some of them in the co...
 Chapter 13.13.3: Find and sketch or graph the sets in the complex plane given by
 Chapter 13.10: What properties of C are similar to those of eX ? Which one is diff...
 Chapter 13.13.4: Are the following functions analytic? [Use (1) or (7).]
 Chapter 13.13.5: Real and Imaginary Parts. Find Re and 1m of ez2
 Chapter 13.13.1: Let':l = 2 + 3i and Z2 = 4  5i. Showing the details of your work. ...
 Chapter 13.13.6: Function Values. Compute (in the form u + iv)
 Chapter 13.13.2: Determine the principal value of the argument.4 ::':: 3;
 Chapter 13.13.7: ll Values of In z. Find all values and graph some of them in the co...
 Chapter 13.13.3: WRITING PROJECT. Sets in the Complex Plane. Extend the part of the ...
 Chapter 13.11: What is the fundamental region of eZ ? Its significance?
 Chapter 13.13.4: (CauchyRiemann equations in polar form) Derive (7) from (1).
 Chapter 13.13.5: Real and Imaginary Parts. Find Re and 1m of
 Chapter 13.13.1: Let':l = 2 + 3i and Z2 = 4  5i. Showing the details of your work. ...
 Chapter 13.13.6: Function Values. Compute (in the form u + iv)
 Chapter 13.13.2: Determine the principal value of the argument.7T2
 Chapter 13.13.7: ll Values of In z. Find all values and graph some of them in the co...
 Chapter 13.13.3: Function Values. Find Re I and 1m f. Also find their values at the ...
 Chapter 13.12: What is an entire function? Give examples.
 Chapter 13.13.4: Are the following functions harmonic? If your answer is yes, find a...
 Chapter 13.13.5: Polar Form. Write in polar form:Vi
 Chapter 13.13.1: Let':l = 2 + 3i and Z2 = 4  5i. Showing the details of your work. ...
 Chapter 13.13.6: Function Values. Compute (in the form u + iv)
 Chapter 13.13.2: Determine the principal value of the argument. 7 ::':: 7;
 Chapter 13.13.7: ll Values of In z. Find all values and graph some of them in the co...
 Chapter 13.13.3: Function Values. Find Re I and 1m f. Also find their values at the ...
 Chapter 13.13: Why is In z much more complicated than In x? Explain from memory.
 Chapter 13.13.4: Are the following functions harmonic? If your answer is yes, find a...
 Chapter 13.13.5: Polar Form. Write in polar form:1 + i
 Chapter 13.13.1: Let':l = 2 + 3i and Z2 = 4  5i. Showing the details of your work. ...
 Chapter 13.13.6: Function Values. Compute (in the form u + iv) sinh (4  3i)
 Chapter 13.13.2: Determine the principal value of the argument.(l + i)12
 Chapter 13.13.7: ll Values of In z. Find all values and graph some of them in the co...
 Chapter 13.13.3: Function Values. Find Re I and 1m f. Also find their values at the ...
 Chapter 13.14: What is the principal value of In z?
 Chapter 13.13.4: Are the following functions harmonic? If your answer is yes, find a...
 Chapter 13.13.5: Polar Form. Write in polar form:V;
 Chapter 13.13.1: Let':l = 2 + 3i and Z2 = 4  5i. Showing the details of your work. ...
 Chapter 13.13.6: Function Values. Compute (in the form u + iv) cosh (4  67Ti)
 Chapter 13.13.2: Determine the principal value of the argument. (9 + 9;)3
 Chapter 13.13.7: ll Values of In z. Find all values and graph some of them in the co...
 Chapter 13.13.3: Function Values. Find Re I and 1m f. Also find their values at the ...
 Chapter 13.15: How is the general power:;c defined? Give examples.
 Chapter 13.13.4: Are the following functions harmonic? If your answer is yes, find a...
 Chapter 13.13.5: Polar Form. Write in polar form: 3 + 4i
 Chapter 13.13.1: Let z = x + iy. Find: Im:3, (1m Z)3
 Chapter 13.13.6: (Real and imaginary parts) Show that sin x cos x Re tan z = =...
 Chapter 13.13.2: Represent in the form x + iy and graph it in the complex plane
 Chapter 13.13.7: ll Values of In z. Find all values and graph some of them in the co...
 Chapter 13.13.3: Continuity. Find out (and give reason) whether .f(z) is continuous ...
 Chapter 13.16: Complex Numbers. Find, in the fonn x + iy. showing the details: (1 ...
 Chapter 13.13.4: Are the following functions harmonic? If your answer is yes, find a...
 Chapter 13.13.5: Polar Form. Write in polar form:9
 Chapter 13.13.1: Let z = x + iy. Find:Re (lIZ)
 Chapter 13.13.6: Equations. Find all solutions of the following equations. cosh z = 0
 Chapter 13.13.2: Represent in the form x + iy and graph it in the complex plane
 Chapter 13.13.7: Show that the set of values of In (i2) differs from the set of valu...
 Chapter 13.13.3: Continuity. Find out (and give reason) whether .f(z) is continuous ...
 Chapter 13.17: Complex Numbers. Find, in the fonn x + iy. showing the details: ( ...
 Chapter 13.13.4: Are the following functions harmonic? If your answer is yes, find a...
 Chapter 13.13.5: Equations. Find all solutions and graph some of them in the complex...
 Chapter 13.13.1: Let z = x + iy. Find: 1m [0 + i)8;;:2]
 Chapter 13.13.6: Equations. Find all solutions of the following equations. sin z = 100
 Chapter 13.13.2: Represent in the form x + iy and graph it in the complex plane
 Chapter 13.13.7: Equations. Solve for z: In z = (2  !i)7T
 Chapter 13.13.3: Continuity. Find out (and give reason) whether .f(z) is continuous ...
 Chapter 13.18: Complex Numbers. Find, in the fonn x + iy. showing the details: 1/(...
 Chapter 13.13.4: Are the following functions harmonic? If your answer is yes, find a...
 Chapter 13.13.5: Equations. Find all solutions and graph some of them in the complex...
 Chapter 13.13.1: Let z = x + iy. Find:Re (1/z2)
 Chapter 13.13.6: Equations. Find all solutions of the following equations.cos Z = 2i
 Chapter 13.13.2: Represent in the form x + iy and graph it in the complex plane
 Chapter 13.13.7: Equations. Solve for z:In z = 0.3 + 0.7;
 Chapter 13.13.3: Continuity. Find out (and give reason) whether .f(z) is continuous ...
 Chapter 13.19: Complex Numbers. Find, in the fonn x + iy. showing the details:(l ...
 Chapter 13.13.4: Are the following functions harmonic? If your answer is yes, find a...
 Chapter 13.13.4: Are the following functions harmonic? If your answer is yes, find a...
 Chapter 13.13.5: Equations. Find all solutions and graph some of them in the complex...
 Chapter 13.13.1: (Laws of addition and multiplication) Derive the following laws for...
 Chapter 13.13.6: Equations. Find all solutions of the following equations.cosh z =  1
 Chapter 13.13.2: Represent in the form x + iy and graph it in the complex plane
 Chapter 13.13.7: Equations. Solve for z:.lnz=e7Ti
 Chapter 13.13.3: Derivative. Differentiate(.:::2  9)/(:::2 + I)
 Chapter 13.20: Complex Numbers. Find, in the fonn x + iy. showing the details:\/5...
 Chapter 13.13.4: Are the following functions harmonic? If your answer is yes, find a...
 Chapter 13.13.5: Equations. Find all solutions and graph some of them in the complex...
 Chapter 13.13.6: Equations. Find all solutions of the following equations.sinh z = 0
 Chapter 13.13.2: Find and graph all roots in the complex plane.
 Chapter 13.13.7: Equations. Solve for z: In z = 2 + ~7Ti
 Chapter 13.13.3: Derivative. Differentiate(:3 + ;)2
 Chapter 13.21: Complex Numbers. Find, in the fonn x + iy. showing the details: (43...
 Chapter 13.13.4: Determine a, b, C such that the given functions are harmonic and fi...
 Chapter 13.13.5: TEAM PROJECT. Further Properties of the Exponential Function. (a) A...
 Chapter 13.13.6: Find all z for which (a) cos z, (b) sin z has real values.
 Chapter 13.13.2: Find and graph all roots in the complex plane.
 Chapter 13.13.7: General Powers. Showing the details of your work, find the principa...
 Chapter 13.13.3: Derivative. Differentiate(3:: + 4i)/( 1.5;:  2)
 Chapter 13.22: Polar Form. Represent in polar form. with the principal argument:
 Chapter 13.13.4: Determine a, b, C such that the given functions are harmonic and fi...
 Chapter 13.13.6: Equations and Inequalities. Using the definitions, prove: cos z is ...
 Chapter 13.13.2: Find and graph all roots in the complex plane.
 Chapter 13.13.7: General Powers. Showing the details of your work, find the principa...
 Chapter 13.13.3: Derivative. Differentiate i/(l  ;::)2
 Chapter 13.23: Polar Form. Represent in polar form. with the principal argument:
 Chapter 13.13.4: Determine a, b, C such that the given functions are harmonic and fi...
 Chapter 13.13.6: Equations and Inequalities. Using the definitions, prove: Isinh yl ...
 Chapter 13.13.2: Find and graph all roots in the complex plane.
 Chapter 13.13.7: General Powers. Showing the details of your work, find the principa...
 Chapter 13.13.3: Derivative. Differentiate::2/(: + ;)
 Chapter 13.24: Polar Form. Represent in polar form. with the principal argument:
 Chapter 13.13.4: (Harmonic conjugate) Show that if II is harmonic and v is a harmoni...
 Chapter 13.13.6: Equations and Inequalities. Using the definitions, prove:sin ZI cos...
 Chapter 13.13.2: Find and graph all roots in the complex plane.
 Chapter 13.13.7: General Powers. Showing the details of your work, find the principa...
 Chapter 13.13.3: CAS PROJECT. Graphing Functions. Find and graph Re f. 1m f. and If ...
 Chapter 13.25: Polar Form. Represent in polar form. with the principal argument:
 Chapter 13.13.4: TEAM PROJECT. Conditions for fez) = COllst. Let f(:;.) be analytic....
 Chapter 13.13.2: TEAM PROJECT. Square Root. (a) Show that w = ~ has the values }\'1 ...
 Chapter 13.13.7: General Powers. Showing the details of your work, find the principa...
 Chapter 13.13.3: TEAM PROJECT. Limit, Continuity, Derivative (a) Limit. Prove that (...
 Chapter 13.26: Polar Form. Represent in polar form. with the principal argument:
 Chapter 13.27: Roots. Find and graph all values of
 Chapter 13.13.4: (Two further formulas for the derivative). Formulas (4). (5), and (...
 Chapter 13.13.2: Solve and graph all solutions, showing the details:
 Chapter 13.13.7: General Powers. Showing the details of your work, find the principa...
 Chapter 13.28: Roots. Find and graph all values of
 Chapter 13.13.4: CAS PROJECT. Equipotential Lines. Write a program for graphing equi...
 Chapter 13.13.2: Solve and graph all solutions, showing the details:
 Chapter 13.13.7: General Powers. Showing the details of your work, find the principa...
 Chapter 13.29: Roots. Find and graph all values of
 Chapter 13.13.2: Solve and graph all solutions, showing the details:
 Chapter 13.13.7: How can you find the answer to Prob. 24 from the answer to Prob. 25?
 Chapter 13.30: Roots. Find and graph all values of
 Chapter 13.13.2: Solve and graph all solutions, showing the details:
 Chapter 13.13.7: TEAM PROJECT. Inverse Trigonometric and Hyperbolic Functions. By de...
 Chapter 13.31: Analytic Functions. Find f(.::) = u(x.y) + ;v(x.y) with 1I or v as ...
 Chapter 13.13.2: CAS PROJECT. Roots of Unity and Their Graphs. Write a program for c...
 Chapter 13.32: Analytic Functions. Find f(.::) = u(x.y) + ;v(x.y) with 1I or v as ...
 Chapter 13.13.2: (Re and 1m) Prove IRe zl ~ Izl, lIm zl ~ Izl
 Chapter 13.33: Analytic Functions. Find f(.::) = u(x.y) + ;v(x.y) with 1I or v as ...
 Chapter 13.13.2: (parallelogram equality) Prove 1::1 + 2212 + 1.:::1  ::;21 2 = 2(h...
 Chapter 13.34: Analytic Functions. Find f(.::) = u(x.y) + ;v(x.y) with 1I or v as ...
 Chapter 13.13.2: (Triangle inequality) Verify (6) for ZI = 4 + 7i. ::2 = 5 + 1;.
 Chapter 13.35: Analytic Functions. Find f(.::) = u(x.y) + ;v(x.y) with 1I or v as ...
 Chapter 13.13.2: (Triangle inequality) Prove (6).
 Chapter 13.36: Harmonic Functions. Are the following functiuns hannonic? If so, fi...
 Chapter 13.37: Harmonic Functions. Are the following functiuns hannonic? If so, fi...
 Chapter 13.38: Harmonic Functions. Are the following functiuns hannonic? If so, fi...
 Chapter 13.39: Harmonic Functions. Are the following functiuns hannonic? If so, fi...
 Chapter 13.40: Special Function Values. Find the values of sin (3 + 47Ti)
 Chapter 13.41: Special Function Values. Find the values ofsinh 47Ti
 Chapter 13.42: Special Function Values. Find the values of cos (57T + 2;)
 Chapter 13.43: Special Function Values. Find the values ofLn CO.8 + 0.6i)
 Chapter 13.44: Special Function Values. Find the values of tan (I + i)
 Chapter 13.45: Special Function Values. Find the values of cosh (I + 7Ti)
Solutions for Chapter Chapter 13: Advanced Engineering Mathematics 9th Edition
Full solutions for Advanced Engineering Mathematics  9th Edition
ISBN: 9780471488859
Solutions for Chapter Chapter 13
Get Full SolutionsChapter Chapter 13 includes 231 full stepbystep solutions. Since 231 problems in chapter Chapter 13 have been answered, more than 12994 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. Advanced Engineering Mathematics was written by Sieva Kozinsky and is associated to the ISBN: 9780471488859. This textbook survival guide was created for the textbook: Advanced Engineering Mathematics, edition: 9.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Factorization
A = L U. If elimination takes A to U without row exchanges, then the lower triangular L with multipliers eij (and eii = 1) brings U back to A.

Fast Fourier Transform (FFT).
A factorization of the Fourier matrix Fn into e = log2 n matrices Si times a permutation. Each Si needs only nl2 multiplications, so Fnx and Fn1c can be computed with ne/2 multiplications. Revolutionary.

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Least squares solution X.
The vector x that minimizes the error lie 112 solves AT Ax = ATb. Then e = b  Ax is orthogonal to all columns of A.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Nullspace N (A)
= All solutions to Ax = O. Dimension n  r = (# columns)  rank.

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.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Rotation matrix
R = [~ CS ] rotates the plane by () and R 1 = RT rotates back by (). Eigenvalues are eiO and eiO , eigenvectors are (1, ±i). c, s = cos (), sin ().

Similar matrices A and B.
Every B = MI AM has the same eigenvalues as A.

Singular matrix A.
A square matrix that has no inverse: det(A) = o.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).
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