A Primer on Complex Numbers

Transcription

1 A Primer on Complex Numbers Intro. to Differentil Equtions April 0, 205 Complex Numbers Definition. A complex number is number tht my be expressed in the form + bi for some rel numbers nd b. The vlue is clled the rel prt of the complex number + ib, nd the vlue b is clled the imginry prt. Here i is specil number stisfying i 2. In other words, i is squre root of. Note tht the rel nd imginry prts of complex number re both rel. We use Re(z) nd Im(z) to denote the rel nd imginry prts of complex number z. In prticulr: Re( + ib), Im( + ib) b. Exercise. Determine the rel nd imginry components of the followng numbers () 3 (b) 7i (c) 3 2i (d) 4 + 7i The set of complex numbers forms field. We cn dd, subtrct, multiply, nd divide (ssuming nonzero denomintor) complex numbers, nd ech of these opertions behves very much in the sme wy s over the rel numbers (commuttive, ssocitive, distributive, etc). Addition/Subtrction: To dd two complex numbers + ib nd c + id, one simply dds the rel nd imginry components: ( + ib) + (c + id) ( + c) + i(b + d). Subtrction is similr: ( + ib) (c + id) ( c) + i(b d).

2 Multipliction: To multiply two complex numbers + ib nd c + id, one simply foils it out : ( + ib)(c + id) c + ibc + id + i 2 bd c bd + i(d + bc). Division of Complex Numbers: The opertion of division for complex numbers is much more interesting. Given complex numbers + ib nd c + id, with the ltter nonzero, we cn clculte ( + ib)/(c + id) vi the following complex conjugtion trick: + ib c + id + ib c + id + ib c + id c id c id c + bd + i(bc d) c 2 + d 2 ( ) ( c + bd bc d + i c 2 + d 2 c 2 + d 2 The bove trick mounts to multiplying nd dividing by the complex conjugte of the denomintor. Definition 2. Let z + ib, be complex number. The complex conjugte of z is denoted z nd given by z ib. In other words the complex conjugte of complex number is the unique complex number with the equl rel prt nd negtive imginry prt. Note tht zz ( + ib)( ib) 2 ib + ib i 2 b 2 2 i 2 b b 2. In prticulr complex number times its complex conjugte is purely rel! Definition 3. Let z + ib be complex number. The the norm (or modulus) of z is z zz 2 + b 2. Exercise 2. Write ech of the following expressions s complex number in the form + ib for nd b rel. () (3 + 7i) + (4 2i) (b) (2 + 3i) ( + 3i) (c) (3 + 6i)( 4i) (d) ( + 2i)/( + 7i) Thinking geometriclly, we cn view complex number + ib s the vector, b in the x, y-plne. Recll tht for vectors in the plne, we hve n lterntive description in terms of the mgnitude of the vector, nd the ngle from the positive x-xis. Note tht the mgnitude of the vector, b is exctly the mgnitude of the complex number + ib. Definition 4. Consider complex number z + ib, nd let θ be the ngle from the positive x-xis to the vector, b in the x, y-plne. Then θ is clled the rgument of z, nd is denoted by Arg(z). 2 ).

3 Since the vector, b is precisely determined by it s mgnitude nd ngle, ech complex number is uniquely determined by its mgnitude nd rgument. Exercise 3. Find the mgnitude nd rgument of ech of the following complex numbers () + i (b) i (c) 3 + 4i 2 Euler s Definition Integrl to our understnding of the complex numbers is the following definition of Euler Definition 5 (Euler). For ny rel number θ, we define the complex number e iθ by e iθ cos(θ) + i sin(θ). Then the big theorem of Euler is the following Theorem. This definition does not brek nything. In other words, the definition grees with the usul lgebric identities of exponentil funtions, ie. e iθ e iφ e i(θ+φ). As well s the usul power series representtion for the exponentil function e iθ (iθ) n. n! n0 As mtter of fct, ny complex number z +ib, if r 2 + b 2 z nd θ Arg(z), then z re iθ. Thus for ny complex number, we cn go bck nd forth between the crtesin + ib form nd the exponentil re iθ form. Exercise 4. Write ech of the following complex numbers in the form re iθ for some rel numbers r nd θ with r 0. () i (b) 2 + 2i (c) 3 + 4i Exercise 5. Write ech of the following complex numbers in the form + ib for some rel vlues nd b.. e iπ 2. 2e iπ/6 3. e i5π/4 4. e ln(2)+iπ 3

4 3 Applictions Proposition. Let θ nd φ be two rel numbers. Then Proof. From Euler s definition, nd lso Now gin from Euler, we clculte: e i(θ+φ) e iθ+iφ e iθ e iφ From this, it follows tht nd lso tht This completes the proof. sin(θ + φ) sin(θ) cos(φ) + cos(θ) sin(φ), cos(θ + φ) cos(θ) cos(φ) sin(θ) sin(φ). sin(θ + φ) Im(e i(θ+φ) ), cos(θ + φ) Re(e i(θ+φ) ). (cos(θ) + i sin(θ))(cos(φ) + i sin(φ)) cos(θ) cos(φ) sin(θ) sin(φ) + i(sin(θ) cos(φ) + cos(θ) sin(φ)). Im(e i(θ+φ) ) sin(θ) cos(φ) + cos(θ) sin(φ) Re(e i(θ+φ) ) cos(θ) cos(φ) sin(θ) sin(φ). Proposition 2. Let nd b be rel numbers. Then e t sin(bt)dt b 2 + b 2 et sin(bt) + C e t b cos(bt)dt 2 + b 2 et sin(bt) + C Proof. Note tht by Euler, e (+ib)t e t+ibt e t e ibt e t (cos(bt) + i sin(bt)) e t cos(bt) + ie t sin(bt). It follows tht nd lso tht Im(e (+ib)t ) e t sin(bt), Re(e (+ib)t ) e t cos(bt). 4

5 Now tking rel or imginry prts commutes with integrtion, nd therefore e t sin(bt)dt Im(e (+ib)t )dt Im( e (+ib)t dt) Im( ( + ib) e(+ib)t ) Where for simplicity we hve left off the usul rbitrry constnt of integrtion. Now using our complex conjugtion trick nd Euler s definition: ( + ib) e(+ib)t ib ( + ib) ib e(+ib)t ( 2 + b ib ) (e t cos(bt) + ie t sin(bt) ) b ( 2 ) b 2 + b 2 et sin(bt) ( ) b + i 2 + b 2 et sin(bt) Thus ( ) ( ) b e t sin(bt)dt Im ( + ib) e(+ib)t 2 + b 2 et sin(bt). nd similrly ( ) ( ) e t b cos(bt)dt Re ( + ib) e(+ib)t 2 + b 2 et sin(bt). Exercise 6. Use Euler to prove the trigonometric identity cos(α + β) + cos(α β) 2 cos(α) cos(β). Exercise 7. The hyperbolic trigonometric functions sinh(x) nd cos(x) re defined by sinh(x) 2 ex 2 e x, nd cosh(x) 2 ex + 2 e x. 5

6 () Use Euler to show tht sinh(x) i sin(ix), cosh(x) cos(ix) This explins their designtion s trigonometric functions. (b) Using (), prove tht d d sinh(x) cosh(x), dx cosh(x) sinh(x) dx One cn obviously lso prove this directly from the definitions, but it s even esier this wy! Exercise 8. Use Euler s definition to clculte the integrl xe 3x cos(2x)dx. 6