COMPLEX ROOTS AND POWERS
14. COMPLEX ROOTS AND POWERS
For real positive numbers, the equation ln a b = b ln a is equivalent to a b =e b ln a . We define complex powers by the same formula with complex a and b. By definition,
a b =e b ln a .
[The case a = e is excluded because we have already defined powers of e by (8.1).] Since ln a is multiple valued (because of the infinite number of values of θ), powers
a b are usually multiple valued, and unless you want just the principal value of ln z or of a b you must use all values of θ. In the following examples we find all values of each complex power and write the answers in the x + iy form.
Example 1. Find all values of i −2i . From Figure 5.2, and equation (13.5) we find ln i =
Ln 1 + i(π/2 ± 2nπ) = i(π/2 ± 2nπ) since Ln 1 = 0. Then, by equation (14.1),
i −2i =e −2i ln i =e −2i·i(π/2±2nπ) =e π±4nπ =e π ,e 5π ,e −3π ,···, where e π = 23.14 · · · . Note the infinite set of values of i −2i , all real! Also read the
end of Section 3, and note that here the final step is not to find sine or cosine of π ± 4nπ; thus, in finding ln i = iθ, we must not write θ in degrees.
Example 2. Find all values of i 1/2 . Using ln i from Example 1 we have i 1/2 =e (1/2) ln i =
e i(π/4+nπ) =e iπ/4 e inπ . Now e inπ = +1 when n is even (Fig. 9.4), and e inπ = −1 when n is odd (Fig. 9.2). Thus,
i 1/2
=±e iπ/4 =± √
1+i
using Figure 5.1. Notice that although ln i has an infinite set of values, we find just two values for i 1/2 as we should for a square root. (Compare the method of Section
10 which is easier for this problem.) Example 3. Find all values of (1 + i) 1−i . Using (14.1) and the value of ln(1 + i) from
Example 2, Section 13, we have
(1 + i) 1−i =e (1−i) ln(1+i) =e (1−i)[Ln 2+i(π/4±2nπ)]
=e Ln
e −i Ln 2 e iπ/4 e ±2nπi e π/4 e ±2nπ
= 2e i(π/4−Ln 2) e π/4 e ±2nπ
(since e ±2nπi = 1)
= 2e π/4 e ±2nπ
[cos(π/4 − Ln
2 ) + i sin(π/4 − Ln
∼ =e ± 2nπ (2.808 + 1.318i).
Now you may be wondering why not just do these problems by computer. The most important point is that it is useful for advanced work to have skill in manip- ulating complex expressions. A second point is that there may be several forms for an answer (see Section 15, Example 2) or there may be many answers (see examples
74 Complex Numbers Chapter 2
above), and your computer may not give you the one you want (see Problem 25). So to obtain needed skills, a good study method is to do problems by hand and compare with computer solutions.
PROBLEMS, SECTION 14
Evaluate each of the following in x + iy form, and compare with a computer solution. √
1. ln(−e)
2. ln(−i)
3. ln(i + 3)
« 4. ln(i − 1)
5. ln(− „1−i 2−i 2) 6. ln √ 2
„1+i « 7. ln
9. (−1) i 1−i
12. i 3+i 13. sin i i 2i/π 14. (2i) 1+i 15. (−1)
√ 3 i « 16. „1+i
2 (i − 1)
17. i+1
18. cos(2i ln i)
i ln 1−i
19. cos(π + i ln 2)
cos[i ln(−1)] »
i ln 23. 1− 2i . Hint: Find 2i first.
24. Show that (a b ) c can have more values than a bc . As examples compare (a)
[(−i) 2+i ] 2−i
and
(−i) (2+i)(2−i) = (−i) 5 ;
(b) (i i ) i and
i −1 .
25. Use a computer to find the three solutions of the equation x 3 −3x−1 = 0. Find a way to show that the solutions can be written as 2 cos(π/9), −2 cos(2π/9), −2 cos(4π/9).
Parts
» CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
» CONDITIONALLY CONVERGENT SERIES
» POWER SERIES; INTERVAL OF CONVERGENCE
» EXPANDING FUNCTIONS IN POWER SERIES
» TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
» ACCURACY OF SERIES APPROXIMATIONS
» COMPLEX POWER SERIES; DISK OF CONVERGENCE
» ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
» POWERS AND ROOTS OF COMPLEX NUMBERS
» THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
» LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
» LINEAR DEPENDENCE AND INDEPENDENCE
» SPECIAL MATRICES AND FORMULAS
» EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
» APPLICATIONS OF DIAGONALIZATION
» A BRIEF INTRODUCTION TO GROUPS
» POWER SERIES IN TWO VARIABLES
» APPROXIMATIONS USING DIFFERENTIALS
» CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
» APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
» MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
» ENDPOINT OR BOUNDARY POINT PROBLEMS
» DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
» APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
» CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
» APPLICATIONS OF VECTOR MULTIPLICATION
» DIRECTIONAL DERIVATIVE; GRADIENT
» SOME OTHER EXPRESSIONS INVOLVING ∇
» GREEN’S THEOREM IN THE PLANE
» THE DIVERGENCE AND THE DIVERGENCE THEOREM
» THE CURL AND STOKES’ THEOREM
» SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
» APPLICATIONS OF FOURIER SERIES
» COMPLEX FORM OF FOURIER SERIES
» LINEAR FIRST-ORDER EQUATIONS
» OTHER METHODS FOR FIRST-ORDER EQUATIONS
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
» OTHER SECOND-ORDER EQUATIONS
» SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
» A BRIEF INTRODUCTION TO GREEN FUNCTIONS
» THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
» SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
» TENSOR NOTATION AND OPERATIONS
» KRONECKER DELTA AND LEVI-CIVITA SYMBOL
» PSEUDOVECTORS AND PSEUDOTENSORS
» VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
» ELLIPTIC INTEGRALS AND FUNCTIONS
» GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
» COMPLETE SETS OF ORTHOGONAL FUNCTIONS
» NORMALIZATION OF THE LEGENDRE POLYNOMIALS
» THE ASSOCIATED LEGENDRE FUNCTIONS
» GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
» THE SECOND SOLUTION OF BESSEL’S EQUATION
» OTHER KINDS OF BESSEL FUNCTIONS
» ORTHOGONALITY OF BESSEL FUNCTIONS
» SERIES SOLUTIONS; FUCHS’S THEOREM
» HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
» LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
» THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHR ¨ODINGER EQUATION
» THE WAVE EQUATION; THE VIBRATING STRING
» STEADY-STATE TEMPERATURE IN A CYLINDER
» VIBRATION OF A CIRCULAR MEMBRANE
» STEADY-STATE TEMPERATURE IN A SPHERE
» INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
» EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
» THE POINT AT INFINITY; RESIDUES AT INFINITY
» SOME APPLICATIONS OF CONFORMAL MAPPING
» THE NORMAL OR GAUSSIAN DISTRIBUTION
» STATISTICS AND EXPERIMENTAL MEASUREMENTS
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