Ch.1:Complex Numbers Ch.1:Complex Numbers L1.5 Powers and Roots L15 Powers and Roots Powers of a Complex Number Roots of a Complex Number We can represent the complex number z in its polar form: The computation of the roots is more complicated than z=rei =r(cos0+isin) powers The n-th power of z is calculated by two steps: Let w=p(cos p+isin)be the m-th roots of Step 1:The n-th power of the modulus:r" z=r(cos0 +isin),so wm =z means Step 2:The n-fold of the angle of inclination:ne p"(cosm+isin mp)=r(cos0+isin) Finally,we get the n-th power of z,namely.2n=r"ein (5) The above rule is valid for both positive and negative integers Eq.(5)means The question arises whether the formula will work for pm =r.cosm cos0,sin mp=sin n=1/m 、Hence,.p=r/m,mp=0+2kr=→p= 0+2k元 n Ch.1:Complex Numbers Ch.1:Complex Numbers L15 Powers and Roots 1.5 Powers and Roots Roots of a Complex Number(Con't) An Example of Finding the Roots Find the Four fourth roots of v2+iv2 When =0,1,2,...,m-1,we get the m distinct roots for 2+iW2=2e Eq-(5)as wk =rl/m cos 8+2k元 +isin 0+2kx） h信 m When k m,m +1,m +2,...,2m-1,the same roots repeat again,… a.你习 Hence,there are only m distinct roots for 21/m 信 迈 迈。信)Ch.1: Complex Numbers 1.5 Powers and Roots Powers of a Complex Number We can represent the complex number z in its polar form: z = reiθ = r(cos θ + isin θ) The n-th power of z is calculated by two steps: Step 1: The n-th power of the modulus: rn Step 2: The n-fold of the angle of inclination: nθ Finally, we get the n-th power of z, namely, zn = rneinθ The above rule is valid for both positive and negative integers The question arises whether the formula will work for n = 1/m Ch.1: Complex Numbers 1.5 Powers and Roots Roots of a Complex Number The computation of the roots is more complicated than powers Let w = ρ(cos ϕ + isin ϕ) be the m-th roots of z = r(cos θ + isin θ), so wm = z means ρm(cos mϕ + isin mϕ) = r(cos θ + isin θ) (5) Eq.(5) means ρm = r, cos mϕ = cos θ, sin mϕ = sin θ Hence, ρ = r1/m, mϕ = θ + 2kπ =⇒ ϕ = θ + 2kπ m Ch.1: Complex Numbers 1.5 Powers and Roots Roots of a Complex Number (Con’t) When k = 0, 1, 2,...,m − 1, we get the m distinct roots for Eq.(5) as wk = r1/m cos θ + 2kπ m + isin θ + 2kπ m  When k = m, m + 1, m + 2,..., 2m − 1, the same roots repeat again,... Hence, there are only m distinct roots for z1/m Ch.1: Complex Numbers 1.5 Powers and Roots An Example of Finding the Roots Find the Four fourth roots of √2 + i√2