1. Sequences and Series in the Complex Plane
Power Series: This is the template for a Power Series: 𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 = 𝑎 0 + 𝑎 1 𝑧− 𝑧 0 + 𝑎 2 𝑧− 𝑧 𝑎 3 𝑧− 𝑧 𝑎 4 𝑧− 𝑧 ⋯ where the coefficients 𝑎 𝑘 are complex constants. This is called a “Power Series centered on 𝒛 𝟎 ”. (Note that the coefficients, 𝑎 𝑘 , are generally not all the same as each other). The power series is said to be centered at 𝑧 0 and the complex number 𝑧 0 is said to be the center of the series. (We note that 𝑧− 𝑧 =1 even when 𝑧= 𝑧 0 .)
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2. Power Series in the Complex Plane
𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 = 𝑎 0 + 𝑎 1 𝑧− 𝑧 0 + 𝑎 2 𝑧− 𝑧 𝑎 3 𝑧− 𝑧 𝑎 4 𝑧− 𝑧 ⋯ Generally, this is not a geometric series because the coefficients, 𝑎 𝑘 , are not (generally) the same as each other. The value of each 𝑎 𝑘 is governed by the rules of the particular series. A geometric series is a special case of a power series. Figuring out the values of the coefficients as functions of “k” will be one our main jobs when working with power series.
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3. Power Series in the Complex Plane
𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 = 𝑎 0 + 𝑎 1 𝑧− 𝑧 0 + 𝑎 2 𝑧− 𝑧 ⋯ The other part of each term in the power series is 𝑧− 𝑧 0 𝑘 where 𝑘 is an integer, 𝑧 0 is a fixed complex number (“a constant”) and 𝑧 is the general complex variable.
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4. Power Series in the Complex Plane
𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 = 𝑎 0 + 𝑎 1 𝑧− 𝑧 0 + 𝑎 2 𝑧− 𝑧 ⋯ An important characteristic of a power series is the rate at which the magnitude of 𝑧− 𝑧 0 𝑘 grows or shrinks as 𝑘 grows. The rate at which 𝑧− 𝑧 0 𝑘 grows is heavily dependent on the choice of 𝑧. If the magnitude of the difference, 𝑧− 𝑧 0 , is too large then the series may “blow up”, i.e., it might diverge. If 𝑧 is selected so that the magnitude, 𝑧− 𝑧 0 , is small enough then the series will converge.
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5. Power Series in the Complex Plane
𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 = 𝑎 0 + 𝑎 1 𝑧− 𝑧 0 + 𝑎 2 𝑧− 𝑧 ⋯ Obviously, if the difference, 𝑧− 𝑧 0 , is chosen to be zero, (that is, if 𝑧= 𝑧 0 ) then the series will converge and it will be equal to 𝑎 0 . Every power series will converge for 𝒛= 𝒛 𝟎 .
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6. Power Series in the Complex Plane
𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 = 𝑎 0 + 𝑎 1 𝑧− 𝑧 0 + 𝑎 2 𝑧− 𝑧 ⋯ It might, on first inspection, seem that if the expression 𝑧− 𝑧 0 >1 then the series must “blow up”, that is, go to ∞. But that’s not necessarily so. If the coefficients, 𝑎 𝑘 , go toward zero faster than the 𝑧− 𝑧 0 𝑘 go toward ∞, then the series might converge. So, we must also consider what happens to 𝑎 𝑘 as 𝑘 goes toward infinity.
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7. Power Series in the Complex Plane
Circle of Convergence: Since we are talking about the sum of an infinite series of complex numbers “going to infinity”, it can go to infinity in any direction. That means the size of the difference, 𝑧− 𝑧 0 , must be smaller than some limit as 𝑧 approaches 𝑧 0 from every direction in order to ensure convergence. So, we talk about the value of the difference 𝑧− 𝑧 0 that is smaller in magnitude than some certain value that will make the series convergent. When 𝑧− 𝑧 0 is small enough, the series will converge. We write it thus: 𝑧− 𝑧 0 <𝑅 Where 𝑅 is called the Radius of Convergence. The circle with radius 𝑅 centered on 𝑧 0 is called the Circle of Convergence.
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8. Sequences and Series in the Complex Plane
Circle of Convergence: Every complex power series is convergent within its circle of convergence which is defined by its radius of convergence, R (R is a non-negative real number) and its center 𝑧 0 . The series converges absolutely for all 𝑧 that fall within the circle 𝑧− 𝑧 0 <𝑅 and diverges for 𝑧− 𝑧 0 >𝑅. The series may or may not be convergent for 𝑧− 𝑧 0 =𝑅 . It could converge for all, some or none of the points on the circle. The radius 𝑅 can be (i) zero, in which case the series only converges for 𝑧 = 𝑧 0 . (ii) a positive real number, in which case it converges for 𝑧− 𝑧 0 <𝑅. (iii) ∞, in which case it converges for all z.
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9. Power Series in the Complex Plane
A commonly used tool for finding the radius of convergence of a power series is the ratio test that is used on infinite series in general. As you surely recall from our work on geometric series, the ratio test works this way: Suppose 𝑘=1 ∞ 𝑧 𝑘 is a series of non-zero complex terms (not necessarily a power series) such that lim 𝑛→∞ 𝑧 𝑛+1 𝑧 𝑛 =𝐿 (i) If 𝐿<1, then the series converges (ii) If 𝐿>1 or 𝐿 →∞, then the series diverges (iii) If 𝐿=1, then the test is inconclusive.
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10. Power Series in the Complex Plane
𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 = 𝑎 0 + 𝑎 1 𝑧− 𝑧 0 + 𝑎 2 𝑧− 𝑧 ⋯ If we apply the ratio test to the general power series (shown above), we get lim 𝑛→∞ 𝑎 𝑛+1 𝑧− 𝑧 0 𝑛+1 𝑎 𝑛 𝑧− 𝑧 0 𝑛 =𝐿 This series is convergent for 𝐿<1.
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11. Power Series in the Complex Plane
So, lim 𝑛→∞ 𝑎 𝑛+1 𝑧− 𝑧 0 𝑛+1 𝑎 𝑛 𝑧− 𝑧 0 𝑛 =𝐿<1 for convergence. lim 𝑛→∞ 𝑎 𝑛+1 𝑧− 𝑧 0 𝑛+1 𝑎 𝑛 𝑧− 𝑧 0 𝑛 = lim 𝑛→∞ 𝑎 𝑛+1 𝑎 𝑛 𝑧− 𝑧 0 𝑛+1 𝑧− 𝑧 0 𝑛 lim 𝑛→∞ 𝑎 𝑛+1 𝑎 𝑛 𝑧− 𝑧 0 = 𝑧− 𝑧 0 lim 𝑛→∞ 𝑎 𝑛+1 𝑎 𝑛
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12. Power Series in the Complex Plane
If 𝑧− 𝑧 0 lim 𝑛→∞ 𝑎 𝑛+1 𝑎 𝑛 <1 then the series is convergent. So, for convergence, we say: 𝑧− 𝑧 0 lim 𝑛→∞ 𝑎 𝑛+1 𝑎 𝑛 <1⇛ 𝑧− 𝑧 0 < lim 𝑛→∞ 𝑎 𝑛 𝑎 𝑛 Recall the definition of Circle of Convergence: 𝑧− 𝑧 0 <𝑅 so we can say 𝑅= lim 𝑛→∞ 𝑎 𝑛 𝑎 𝑛+1
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13. Sequences and Series in the Complex Plane
Example (Circle of Convergence): Consider the series 𝑘=1 ∞ 𝑧 𝑘+1 𝑘 Where does the series converge? Using the 𝒓𝒂𝒕𝒊𝒐 𝒕𝒆𝒔𝒕, 𝑅= lim 𝑛→∞ 𝑧 𝑛+2 𝑛 𝑧 𝑛+1 𝑛 = lim 𝑛→∞ 𝑛 𝑛+1 ∙ 𝑧 𝑛+2 𝑧 𝑛+1 = lim 𝑛→∞ 𝑛 𝑛+1 𝑧 = 𝑧 If this limit is less than one then the series converges and if the limit is greater than one then the series diverges. So the series converges absolutely for 𝑧 <1 and the radius of convergence is 𝑅=1.
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14. Power Series in the Complex Plane
Example (Finding the Circle of Convergence): Consider the series 𝑘=0 ∞ −2𝑖 𝑘 𝑧−4−3𝑖 𝑘 Where does the series converge? This is a power series centered on 𝑧 0 = 4+3𝑖 . The 𝑎 𝑘 coefficient for each term is 𝑎 𝑘 = 1 1−2𝑖 𝑘+1 . So, using the rule for finding the radius of convergence, we have: 𝑅= lim 𝑛→∞ 𝑎 𝑛 𝑎 𝑛+1 = lim 𝑛→∞ −2𝑖 𝑛 −2𝑖 𝑛 = lim 𝑛→∞ 1−2𝑖 = 5 Therefore the “Circle of Convergence” (C of C) is 𝑧−4−3𝑖 = 5
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15. 𝑘=0 ∞ 1 1−2𝑖 𝑘+1 𝑧−4−3𝑖 𝑘 𝐶 𝑜𝑓 𝐶: 𝑧−4−3𝑖 = 5
Power Series in the Complex Plane
𝑘=0 ∞ −2𝑖 𝑘 𝑧−4−3𝑖 𝑘 𝐶 𝑜𝑓 𝐶: 𝑧−4−3𝑖 = 5
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16. Power Series in the Complex Plane
Example (Finding the Circle of Convergence): Consider the series 𝑘=0 ∞ 𝑧 1+3𝑖 +3−𝑖 𝑘 Where does the series converge? First, we must work on it to determine if it is a power series. We must get it in the right form. Rework the expression in the brackets (this is kind of tricky): 𝑧 1+3𝑖 +3−𝑖=𝑧 1+3𝑖 −𝑖 3𝑖+1 = 1+3𝑖 𝑧−𝑖
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17. Power Series in the Complex Plane
So, now we have: 𝑘=0 ∞ 𝑖 𝑧−𝑖 𝑘 = 𝑘=0 ∞ 𝑖 𝑘 𝑧−𝑖 𝑘 This is a power series with 𝑎 𝑘 = 1+3𝑖 𝑘 centered on 𝑧 = 𝑖. The radius of convergence is given by: 𝑅= lim 𝑛→∞ 𝑖 𝑛 𝑖 𝑛+1 = 𝑖 −1 = The circle of convergence is 𝑧−𝑖 =
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18. Power Series in the Complex Plane
Continuity: Because the power series resolves to a particular value (“is convergent”) at all points within its circle of convergence, it fits the definition of a continuous function in that region. We state (quote): 𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 (a power series) represents a continuous function 𝒇 within its circle of convergence 𝑧− 𝑧 0 =𝑅, 𝑅>0. (end quote)
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19. Power Series in the Complex Plane
Differentiation: Because a power series is really a polynomial with an infinity of terms, each term comprised of a numerical constant coefficient and a binomial of (𝑧− 𝑧 0 ) raised to an integer power, the whole power series can be differentiated using the same simple rules for taking the derivative of a polynomial. Formally: “A power series 𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 can be differentiated, term by term, within its circle of convergence 𝑧− 𝑧 0 =𝑅, 𝑅∈ℝ>0.”
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20. Power Series in the Complex Plane
Integration: For the same reasons that the power series can be differentiated, it can also be integrated rather easily as a polynomial. The path of integration (whether open or closed) must lie entirely within the circle of convergence. Formally: “A power series 𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 can be integrated, term by term, within its circle of convergence 𝑧− 𝑧 0 =𝑅, 𝑅∈ℝ>0 for every contour lying entirely within the circle of convergence.”
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21. Power Series in the Complex Plane
As stated above in reference to continuity: “A power series 𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 represents a continuous function 𝑓 within its circle of convergence 𝑧− 𝑧 0 =𝑅, 𝑅∈ℝ≥0.” So, the power series represents a differentiable function within the finite circle of convergence: 𝑓 𝑧 = 𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 = 𝑎 0 + 𝑎 1 𝑧− 𝑧 0 + 𝑎 2 𝑧− 𝑧 𝑎 3 𝑧− 𝑧 ⋯ Each term is easily differentiable.
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22. Power Series in the Complex Plane
𝑓 𝑧 = 𝑘=0 ∞ 𝑎 𝑘 𝑧− 𝑧 0 𝑘 = 𝑎 0 + 𝑎 1 𝑧− 𝑧 0 + 𝑎 2 𝑧− 𝑧 𝑎 3 𝑧− 𝑧 ⋯ Each term is easily differentiable at every point within the circle of convergence. The region within (but not on) the circle of convergence is an open and connected set. Therefore, the region is a domain. Because the power series is differentiable at all points within this domain, we can say that it is analytic within the circle of convergence. Therefore, any function that can be represented as a power series with a finite domain within its circle of convergence is analytic at all points within that 𝐶 𝑜𝑓 𝐶. This will be of great use when develop the Taylor Series.
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