Complex Analysis
Sequences and Limits
Draft for Information Only Content Sequences and Limits source/reference: Sequences and LimitsSequencesConsider the following sequences of complex numbers. 1, 1/2, 1/3, 1/4, 1/5, 1/6,…1/n→s 𝑖, 𝑖/2, 𝑖/3, 𝑖/4, 𝑖/5, 𝑖/6,…𝑖/n→s 𝑖, 𝑖/2, 𝑖/3, 𝑖/4, 𝑖/5, 𝑖/6,…𝑖ⁿ/n→s Unlike sequences of real number, a complex number sequence {sₙ} converges to a limit s if the sequence eventually lies in any (every so small) disk centered at s. By definition. A sequence {sₙ} of complex numbers converges to s∊ℂ if for every ε>0 there exists an index N≥1 such that ss<ε for all n>N. That is lim
n→∞sₙ=s
For example, lim n→∞1 n=0 lim n→∞1 np=0 for any 0<p<∞ lim n→∞c np=0 for any c∊ℂ, 0<p<∞ lim n→∞qn=0 for 0<q<1 lim n→∞zn=0 for z<1 lim
n→∞ⁿ√10=1
lim
n→∞ⁿ√n=1
Rules for Limits
For examples n n+1= 1 1+1 n→1 as n→∞ 3n²+5 𝑖n²+2𝑖n1=3+5 n² 𝑖+2𝑖 n1 n²→3 𝑖=3𝑖 as n→∞ n² n+1= n 1+1 n→n as n→∞, not bounded 3n+5 𝑖n²+2𝑖n1=3 n+5 n² 𝑖+2𝑖 n1 n²→0 𝑖=0 as n→∞ Convergence of Complex Number SequencesA sequence of complex numbers, {sₙ}, converges to 0 if and only if the sequence {sₙ} of absolute values converges to 0. And a sequence of complex numbers, {sₙ}, with sₙ=xₙ+𝑖yₙ, converges to s=x+𝑖y if and only if xₙ→x and yₙ→y as n→∞. For example {𝑖ⁿ n}=𝑖,1 2,𝑖 3,1 4,𝑖 5,1 6,…→0 as n→∞ Facts about Sequence of Real NumbersBy Squeeze Theorem, suppose that {rₙ}, {sₙ}, and {tₙ} are sequences of real numbers such that rₙ≤sₙ≤tₙ for all n. If both sequences {rₙ} and {tₙ} converge to the same limit, L, then the sequence {sₙ} has no choice but to converge to the limit L as well. By theorem. A bounded, monotone sequence of real numbers converges. For example, Complex Number Sequences, {𝑖ⁿ 𝑖ⁿ n=𝑖ⁿ n=1 n→0 as n→∞. Thus lim n→∞𝑖ⁿ n=0 Let 𝑖ⁿ n=xₙ+𝑖yₙ, ⇒xₙ={0, n=odd 1/n, n=4k, 1/n, n=4k+2, yₙ={0, n=even 1/n, n=4k+1, 1/n, n=4k+3 Since 1/n≤xₙ≤1/n, and 1/n≤yₙ≤1/n for all n, the Squeeze theorem implies that lim n→∞xₙ=0 and lim n→∞yₙ=0, hence lim n→∞𝑖ⁿ n=0 Limits of Complex FunctionsBy definition. The complexvalued function f(z) has limit L as z→z₀ if the values of f(z) are near L as z→z. That is lim
z→z₀f(z)=L if for all ε>0 there exists δ>0 such that f(z)L<ε whenever 0<zz₀<δ.
Where f(z) needs to be defined near z₀ for this definition to make sense, but is not necessary at z₀.
For example, f(z)=z²1 z1,z≠1. Then lim z→1f(z)=lim z→1z²1 z1=lim z→1(z1)(z+1) z1=lim z→1z+1=2 Let f(z)=Arg z. Then: lim z→𝑖Arg z=π 2 lim
z→1Arg z=0
lim
z→1Arg z=does not exist. since π<Arg z≤π
Facts about Limits of Complex Functions
ContinuityBy definition. The function f is continuous at z₀, if f(z)→f(z₀) as z→z₀. f is defined at z₀. f has a limit as z→z₀. The limit equals f(z₀). Examples: constant functions f(z)=z polynomials f(z)=z f(z)=P(z)/q(z) wherever q(z)≠0 (p and q are polynomials). ©sideway ID: 190300018 Last Updated: 2019/3/18 Revision: 
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