Ch.1:Complex Numbers Ch.1:Complex Numbers L1.4 The Complex Exponential L14 The Complex Exponential Euler's Equation (Cont'd) Euler's Equation(Cont'd) By replacing y with iy in Eq.(1),we get the Taylor's By separating the real part and imaginary part of e,Eq.(2) expansion of ety as follows can be rewritten as =1++g+g+g+罗 + (2) --苦+若-+(-苦+黄-… We know the identities Note that the real part and the imaginary part of the above =i,2=-1,3=-i,4=1 are just the Taylor's expansions of cosy and siny,respectively 泸=i,=-1,7=-i,8=1, (3) Hence,We arrive at the famous Euler's equation as follows eiv cosy+isiny Hence,we deduce that i=im+4 is periodical function with By using the Euler's equation,we have the definition of a period 4 complex exponential function:e*:=e*(cosy+isiny) Ch.1:Complex Numbers Ch.1:Complex Numbers L14 The Complex Exponential 14 The Complex Exponential Comments to Euler's Equation Application of Complex Exponential Since leiu=cosy+isinyl Vcos2y+sin2y=1,ei is a 2—；sin0=3e0-e0-e-0 ·os9=Re0=e0+e0 2i vector which locates on the circle of radius 1 about origin Multiplication of two complex numbers: y is the angle of inclination of the vector eiy,measured 2a2=(r1ei01)(r2e）=(r12)e0+) positively in a counterclockwise sense from the positive real Division of two complex numbers: axis Recall that any complex number z can be written as the polar 号-受- T2 form:z=r(cos0+isin) Complex Conjugate:=re-i0 Euler's equation enables us to write it in another form: De Moivre's Formula:(cos0+isin0)n=cosne +isin no 2=rei leleiargs Q:Does this formula hold for arbitrary integers n(positive or negative)?Ch.1: Complex Numbers 1.4 The Complex Exponential Euler’s Equation (Cont’d) By replacing y with iy in Eq. (1), we get the Taylor’s expansion of eiy as follows eiy =1+ iy + (iy)2 2! + (iy)3 3! + (iy)4 4! + (iy)5 5! + ... (2) We know the identities i1 = i, i2 = −1, i3 = −i, i4 = 1 i5 = i, i6 = −1, i7 = −i, i8 = 1,... (3) Hence, we deduce that in = in+4 is periodical function with period 4 Ch.1: Complex Numbers 1.4 The Complex Exponential Euler’s Equation (Cont’d) By separating the real part and imaginary part of eiy, Eq. (2) can be rewritten as eiy = 1 − y22! + y44! − ... + i y − y33! + y55! − ... (4) Note that the real part and the imaginary part of the above are just the Taylor’s expansions of cos y and sin y, respectively Hence, We arrive at the famous Euler’s equation as follows eiy = cos y + isin y By using the Euler’s equation, we have the definition of a complex exponential function: ez := ex(cos y + isin y) Ch.1: Complex Numbers 1.4 The Complex Exponential Comments to Euler’s Equation Since |eiy| = | cos y + isin y| = cos2 y + sin2 y = 1, eiy is a vector which locates on the circle of radius 1 about origin y is the angle of inclination of the vector eiy, measured positively in a counterclockwise sense from the positive real axis Recall that any complex number z can be written as the polar form: z = r(cos θ + isin θ) Euler’s equation enables us to write it in another form: z = reiθ = |z|ei arg z Ch.1: Complex Numbers 1.4 The Complex Exponential Application of Complex Exponential cos θ = eiθ = eiθ + e−iθ 2 ; sin θ = eiθ = eiθ − e−iθ 2i Multiplication of two complex numbers: z1z2 = r1eiθ1 r2eiθ2 = (r1r2)ei(θ1+θ2) Division of two complex numbers: z1 z2 = r1eiθ1 r2eiθ2 = r1 r2 ei(θ1−θ2) Complex Conjugate: z = re−iθ De Moivre’s Formula: (cos θ + isin θ)n = cos nθ + isin nθ Q: Does this formula hold for arbitrary integers n (positive or negative)?