Euler’s Formula, Polar Representation

Euler’s Formula, Polar Representation 1. The Complex Plane Complex numbers are represented geometrically by points in the plane: the number a + ib is represented by the point ( a, b) in Cartesian coordinates. When the points of the plane are thought of as representing complex num­ bers in this way, the plane is called the complex plane. By switching to polar coordinates, we can write any non-zero complex number in an alternative form. Letting as usual x = r cos(θ ), y = r sin(θ ) we get the polar form for a non-zero complex number: assuming x + iy �= 0, (1) x + iy = r (cos(θ ) + i sin(θ )). When the complex number is written in polar form, � r = | x + iy| = x2 + y2 . (absolute value, modulus). We call θ the polar angle or the argument of x + iy. In symbols, one sometimes sees: θ = arg( x + iy). (polar angle, argument). The absolute value is uniquely determined by x + iy but the polar angle is not, since it can be increased by any integer multiple of 2π. (The complex number 0 has no polar angle.) To make θ unique, one can specify 0 ≤ θ < 2π. (principal value). This so-called principal value of the angle is sometimes indicated by writing Arg( x + iy). For example, Arg(−1) = π, arg(−1) = ±π, ±3π, ±5π, · · · Changing between Cartesian and polar representation of a complex number is essentially the same as changing between Cartesian and polar coordinates: the same equations are used and the same triangle appears in the plane. The figure below shows this. (You will learn what eiθ means in the next section.) Euler’s Formula, Polar Representation OCW 18.03SC y z = x + iy = reiθ r = |z| y θ x x z̄ = x − iy = re−iθ Fig 1. The complex plane. √ Example 1. Give the polar form for: −i, 1 + i, 1 − i, −1 + i 3. Solution. √ −i = i sin(3π/2) 1 + i = 2(cos(π/4) + i sin(π/4)) √ √ −1 + i 3 = 2(cos(2π/3) + i sin(2π/3)) 1 − i = 2(cos(−π/4) + i sin(−π/4)). 2. Euler’s Formula The abbreviation cis θ is sometimes used for cos(θ ) + i sin(θ ); for stu­ dents of science and engineering, however, it is important to get used to the exponential form for this expression: eiθ = cos(θ ) + i sin(θ ) Euler’s formula. (2) Equation (2) should be regarded as the definition of the exponential of an imaginary power. A good justification for it is found in the infinite series: et = 1 + t t2 t3 + + +··· 1! 2! 3! If we substitute iθ for t in the series and collect the real and imaginary parts of the sum (remembering that i2 = −1, i3 = −i, i4 = 1, i5 = i, and so on), we get: � � � � θ2 θ4 θ3 θ5 iθ e = 1− + −··· +i θ − + −··· 2! 4! 3! 5! = cos(θ ) + i sin(θ ) 2 Euler’s Formula, Polar Representation OCW 18.03SC in view of the infinite series representations for cos(θ ) and sin(θ ). Since we only know that the series expansion for et is valid when t is a real number, the above argument is only suggestive — it is not a proof of (2). What it shows is that Euler’s formula (2) is formally compatible with the series expansions for the exponential, sine, and cosine functions. 3. Polar Representation Using the complex exponential, the polar representation (1) is written: x + iy = reiθ . (3) The most important reason for polar representation is that multiplica­ tion of complex numbers is particularly simple when they are written in polar form. Indeed, by using Euler’s formula (2) and the trigonometric addition formulas, it is not hard to show that eiθ1 eiθ2 = ei(θ1 +θ2 ) . (4) This gives another justification for the definition (2) — the complex expo­ nential follow the same exponential addition rules as the real exponential. The law (4) leads to the simple rules for multiplying and dividing complex numbers written in polar form: Multiplication Rule r1 eiθ1 · r2 eiθ2 = r1 r2 ei(θ1 +θ2 ) . (5) To multiply two complex numbers, you multiply the absolute values and add the angles. Reciprocal Rule 1 1 = e−iθ ; iθ r re Division Rule (6) r1 eiθ1 r = 1 e i ( θ1 − θ2 ) . (7) iθ 2 r2 r2 e To divide by a complex number, divide by its absolute value and subtract its angle. The reciprocal rule (6) follows from (5), which shows that 1 −iθ e · reiθ = 1. r 3 Euler’s Formula, Polar Representation OCW 18.03SC Using (5), we can raise x + iy to a positive integer power by first using x + iy = reiθ : ( x + iy)n = r n einθ ; (8) DeMoivre’s formula: The special case when r = 1 is called DeMoivre’s Formula: (cos(θ ) + i sin(θ ))n = cos(nθ ) + i sin(nθ ). (9) Example 2. Express: a) (1 + i )6 in Cartesian form; √ 1+i 3 b) √ in polar form. 3+i Solution. a) Change to polar form, use (8), then change back to Cartesian form: √ √ (1 + i )6 = ( 2eiπ/4 )6 = ( 2)6 ei6π/4 = 8ei3π/2 = −8i. √ 1+i 3 2eiπ/3 b) Changing to polar form, √ = iπ/6 = eiπ/6 , using the division 2e 3+i rule (7). You can check the answer to (a) by applying the binomial theorem to (1 + i )6 and collecting the real and imaginary parts; to (b) by doing the division in the Cartesian form then converting the answer to polar form. 3.1. Combining pure oscillations of the same frequency. The equation which does this is widely used in physics and engineer­ ing; it can be expressed using complex numbers: where a + bi = Aeiφ ; a cos(λt) + b sin(λt) = A cos(λt − φ), √ in other words, A = a2 + b2 , φ = tan−1 (b/a). To prove (10), we have: a cos(λt) + b sin(λt) = Re (( a − bi ) · (cos(λt) + i sin(λt))) = Re( Ae−iφ · eiλt ) = Re( Aei(λt−φ) ) = A cos(λt − φ). 4 (10)