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Complex Algebra Imaginary Numbers: Multiples of –1 = i Properties of i : i • i = i^2 = – 1 and i ÷ i = i / i = + Complex Numbers ( C ): composed of a real number ( R ) and an imaginary number For complex number Z = X + i Y, where X and Y are real numbers, we can define the "operators" Re(Z) = X and Im(Z) = Y. Agrand Diagram: The real component of a complex number is taken as the x - component of an ordered pair, and the imaginary part is the y - component. The complex number is the point on the graph corresponding to ( x , y ): ( x (^) 0 , y 0 (^) ) = z 0 x 0 0 y y = Im(z) x = Re(z) Algebra of Complex Numbers: For most of the discussion we will use two generic complex numbers: z = (x + i y) and w = (a + i b) Addition and Subtraction: z + w = (x+ i y) + (a+ i b) = (x+a) + i (y+b) z – w = (x+ i y) – (a+ i b) = (x–a) + i (y–b) Multiplication: z • w = (x+ i y)•(a+ i b) = x•(a+ i b) + i y•(a+ i b) = xa + i xb + i ya – yb = (xa–yb) + i (xb+ya) z • z = z^2 = (x+ i y)•(x+ i y) = (x^2 – y^2 ) + i (x^2 +y^2 ) = x^2 •(1+ i ) + y^2 •( i – 1) Definition - Complex Conjugate: If Z=X+ i Y is a complex number, then its complex conjugate, Z^ is Z^ = X– i Y (i.e. i is replaced by – i ). z • z^ = (x+ i y)•(x– i y) = x^2 – i xy + i yx + y^2 = x^2 + y^2 (A real number) NOTE: (Z•W)^ = Z•W
Complex Algebra Definition - Modulus: The modulus of a complex number |Z| = |X+ i Y| is given by Z = Z•Z^ 1/ = X^2 + Y^2 NOTE: The modulus is the distance to point z from the origin in an Agrand diagram. Division: z w =^ z w •^ w w*^
z•w* w 2
(x+ i y)•(a– i b) (a^2 + b^2 )
(xa+yb) – i (xb–ya) (a^2 + b^2 )
(xa+yb) + i (ya–xb) (a^2 + b^2 ) NOTE: This is the same idea as rationalizing a denominator! Fundamental Theorem of Algebra: A complex polynomial of order n has exactly n complex roots, some of which are NON-unique. Polar Representations of Complex Numbers: Especially useful when thinking in terms of Agrand diagrams: ( x (^) 0 , y 0 (^) ) = z 0 x 0 0 y y = Im(z) x = Re(z) = (r ,θ 0 ) 0 θ 0 r 0 Based on trigonometry: x = r cos(θ) and y = r sin(θ) z = (x + i y) = r (cos(θ) + i sin(θ)) |z| = [r^2 (cos^2 (θ) + sin^2 (θ))]1/2^ = r Definition - Argument (Arg): The argument of a complex number, Arg(Z), is the angle that represents it in the polar representation of the number: Arg(Z) = Arg(X+ i Y) = Arg(R(cos(θ)+ i sin(θ))) = θ. Review - Trigonometric Identities: cos(–α) = +cos(α) EVEN FUNCTION sin(–α) = – sin(α) ODD FUNCTION sin(α+β) = sin(α)cos(β) + cos(α)sin(β) sin(α–β) = sin(α)cos(β) – cos(α)sin(β) cos(α+β) = cos(α)cos(β) – sin(α)sin(β)
Complex Algebra = r^2 + ρ^2 + rρ{cos(θ–φ) + i sin(θ–φ) + cos(–(θ–φ)) + i sin(–(θ–φ))} = r^2 + ρ^2 + rρ{cos(θ–φ) + i sin(θ–φ) + cos(θ–φ) – i sin(θ–φ)} = r^2 + ρ^2 + 2rρ cos(θ–φ) = δ^2 In other words, δ = [r^2 + ρ^2 + 2rρ cos(θ–φ)]1/ Re(q) = Re(z+w) = Re(z) + Re(w) = δ cos(ψ) = r cos(θ) + ρ cos(φ) Im(q) = Im(z+w) = Im(z) + Im(w) = δ sin(ψ) = r sin(θ) + ρ sin(φ) Definition - Demoivre's Theorem: If a complex number z = (x + i y) = r (cos(θ) + i sin(θ)) is raised to a real power n , the expression can be rewritten: z n^ = r n^ (cos(θ) + i sin(θ)) n^ = r n^ (cos( n θ) + i sin( n θ)) note: Since n is any real number, it can be fractional and/or negative. Example: z–^1 = 1/z = r–^1 (cos(–θ) + i sin(–θ)) = (1/r) (cos(θ) – i sin(θ)) Note, if r=1, then z–^1 = z•. Therefore: cos(θ) = z^ +^ z
- 1 2 and sin(θ) = z – z– 1 2 i This can be used to "prove" Demoivre's theorem, since: cos n^ (θ) = (z + z– 1) n 2 n^
2 n^ n! n! z n^ + n! ( n –1)!1! z n – 2^ + n! ( n –2)!2! z n – 4^ + ... + n! n!
z n = 1 2 n – 1 n! n! cos(nθ) + n! ( n –1)!1! cos({n–2}θ) + n! ( n –2)!2! cos({n–4}θ) + ... + 1 sin n^ (θ) = (z – z– 1) n i n^ 2 n^
i n^ 2 n^ n! n! z n^ – n! ( n –1)!1! z n – 2^ + n! ( n –2)!2! z n – 4^ – ... – n! n!
z n = 1 i n – 1 2 n – 1 n! n! sin(nθ) – n! ( n –1)!1! sin({n–2}θ) + n! ( n –2)!2! sin({n–4}θ) – ... + 0 Note: these were calculated using the polynomial expansion Complex roots of the number 1:.
Complex Algebra
Vector Algebra s • ( a + b ) = s • a + s • b Magnitude of a Vector | a | = ( ax^2 + ay^2 + az^2 )1/ Arbitrary Unit Vector: ê b = b /| b | Addition: a + b = ( ax + bx ) î + ( ay + by ) ˆj + ( az + bz ) ˆk a b b a + b Multiplication: Several different types: "Dot" (Scalar) Product: a • b = | a | | b | cos(θ) a b θ In terms of individual terms: a • b = ax î • b + ay ˆj • b + az ˆk • b = ax î • ( bx î + by ˆj + bz ˆk ) + ... = ( axbx ) î • î + ( axby ) î • ˆj + ( axbz ) î • ˆk + .... But, if we look at the individual unit vector dot products we see:
Vector Algebra î • î = (1)(1) cos(0) = 1 î • ˆj = î • ˆk = (1)(1) cos(π/2=90˚) = 0 Therefore: a • b = ( axbx ) + ( ayby ) + ( azbz ) = b • a Note: The dot product can be used as a way of finding the angle between two vectors: θ = cos– 1^ a • b a b = arccos a • b a b Polar Representations of Vectors: This is similar to the Agrand diagrams: 2 - D Projections: a = ax î + ay ˆj = | a | cos(α) î + | a | sin(α) ˆj a x a y y x a α ax is said to be the "projection of vector a onto the x-axis", and similarly for ay. Note: The dot product can be considered a projection of one vector onto the other: projection of b onto a = | b | cos(θ) = [ a • b ]/| a | Examples of physical quantities that are dot products (projections): work = F • s = ∫ F • d s , where F is the force in the direction of the displacement, s. voltage = – ∫ E • d s = – ∫ ( F /q) • d s , where E is the electric field and q is a charge. 3 - D Projections: Spherical-Polar coordinates R = x î + y ˆj + z ˆk = | R | sin(θ) cos(φ) î + | R | sin(θ) sin(φ) ˆj + | R | cos(θ) ˆk R θ z y x φ
Vector Algebra
