COMPLEX POWER SERIES; DISK OF CONVERGENCE
7. COMPLEX POWER SERIES; DISK OF CONVERGENCE
In Chapter 1 we considered series of powers of x, n x n . We are now interested in series of powers of z,
where z = x + iy, and the a n are complex numbers. [Notice that (7.1) includes real series as a special case since z = x if y = 0.] Here are some examples.
(7.2a) 1−z+
(iz) 2 (iz) 3 z 2 iz 3 (7.2b)
3! + · · · = 1 + iz − 2! − 3! +···, ∞ (z + 1 − i) n (7.2c)
Let us use the ratio test to find for what z these series are absolutely convergent. For (7.2a), we have
z·n
ρ = lim
n→∞ n+1 = |z|.
The series converges if ρ < 1, that is, if |z| < 1, or
2 +y 2 < 1. This is the interior of a disk of radius 1 with center at the origin in the complex plane. This disk is called
the disk of convergence of the infinite series and the radius of the disk is called the radius of convergence. The disk of con- vergence replaces the interval of convergence which we had for real series. In fact (see Figure 7.1), the interval of con- vergence for the series
n /n is just the interval (−1, 1) on the x axis contained within the disk of convergence of
Figure 7.1 n /n, as it must be since x is the value of z when y = 0. For this reason we sometimes speak of the radius of convergence of a power series even though we are considering only real values of z. (Also see Chapter 14, Equations (2.5) and (2.6) and Figure 2.4.)
Next consider series (7.2b); here we have
(iz) n+1
This is an example of a series which converges for all values of z. For series (7.2c), we have
ρ = lim
(z + 1 − i)
z+1−i
n→∞ 3 (n + 1)
Thus, this series converges for |z + 1 − i| < 3, or |z − (−1 + i)| < 3.
This is the interior of a disk (Figure 7.2) of radius 3 and Figure 7.2 center at z = −1 + i (see Problem 5.65).
Section 7 Complex Power Series; Disk of Convergence
Just as for real series, if ρ > 1, the series diverges (Problem 6.14). For ρ = 1 (that is, on the boundary of the disk of convergence) the series may either converge or diverge. It may be difficult to find out which and we shall not in general need to consider the question.
The four theorems about power series (Chapter 1, Section 11) are true also for complex series (replace interval by disk of convergence). Also we can now state for Theorem 2 what the disk of convergence is for the quotient of two series of powers of z. Assume to start with that any common factor z has been cancelled. Let r 1
and r 2 be the radii of convergence of the numerator and denominator series. Find the closest point to the origin in the complex plane where the denominator is zero; call the distance from the origin to this point s. Then the quotient series converges
at least inside the smallest of the three disks of radii r 1 ,r 2 , and s, with center at the origin. (See Chapter 14, Section 2.)
Example. Find the disk of convergence of the Maclaurin series for (sin z)/[z(1 + z 2 )]. We shall soon see that the series for sin z has the same form as the real series
for sin x in Chapter 1. Using this fact we find (Problem 17)
sin z
7z 2 47z 4 5923z 6
6 40 − 5040 +···. From (7.3) we can’t find the radius of convergence, but let’s use the theorem above.
Let the numerator series be (sin z)/z. By ratio test, the series for (sin z)/z converges for all z (if you like, r 1 = ∞). There is no r 2 since the denominator is not an infinite series. The denominator 1 + z 2 is zero when z = ±i, so s = 1. Then the series (7.3) converges inside a disk of radius 1 with center at the origin.
PROBLEMS, SECTION 7
Find the disk of convergence for each of the following complex power series.
1. e z z =1+z+ 2 + z 3
6. n 2 (3iz) n
7. (−1) z
n 2n
(2n)! X ∞
X ∞ (iz) n
X ∞ (n!) 3 z n
X ∞ (n!) 2 z n
n(n + 1)(z − 2i) 2n 15.
14. (z − 2 + i)
16. 2 (z + i − 3)
2 n=1
n=0
n=0
60 Complex Numbers Chapter 2
17. Verify the series in (7.3) by computer. Also show that it can be written in the form
(−1) n z 2n X 1 .
k=0 (2k + 1)!
n=0
Use this form to show by ratio test that the series converges in the disk |z| < 1.
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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