Complex Numbers

Math 441, Summer 2018 A.J. Hildebrand

Basic Deﬁnitions and Properties

• A complex number is a number of the form z = a + ib , where a, b are real numbers and i is the imaginary unit, the square root of −1, i.e., i satisﬁes i2 = −1 .

• The real part and imaginary part of a complex number z = a + ib are deﬁned as Re(z) = a and Im(z) = b .

• The modulus of z (or absolute value or magnitude) a complex number z = a + ib is deﬁned p as |z| = a2 + b2 .

• Geometric representation: A complex number z = a + ib can be thought of as point (a, b) in the plane. The real and imaginary parts of z represent the coordinates this point, and the absolute value represents the distance of this point to the origin. • Complex conjugate: Given z = a + ib, the complex number z = a − ib is called the complex conjugate of z. Geometrically, z is the reﬂection of z by the positive real axis.

• Euler’s Formula: eiθ = cos θ + i sin θ This famous formula connects trig functions with exponentials via complex numbers. It is this connection that makes complex numbers such a useful tool in many problems. The complex exponential eiθ in this formula satisﬁes all the usual properties of an exponential. A consequence of Euler’s formula are the following relations for sin and cos: 1 1 cos θ = eiθ + e−iθ , sin θ = eiθ − e−iθ 2 2i • Polar representation: If z is a nonzero complex number, it can be written in the form z = reiθ = r(cos θ + i sin θ) , where r = |z| and θ is the “angle”, or argument, of z. In the above geometric representation of z, r is the distance of z to the origin, and√θ is the angle formed by the segment Oz and the positive real axis. Thus, if z = a + ib, then r = a2 + b2 and tan θ = b/a. • Arithmetic with complex numbers: Complex numbers can be added, subtracted, multiplied, and scaled (i.e., multiplied with a real number), using the usual rules along with the relation i2 = −1 to simplify the resulting expressions and put them in standard form a+ib. Division works similarly, except that one needs to “rationalize” the denominator by multiplying with its complex conjugate. The exponential of a complex number can be calculated using the usual rules for exponentials (e.g., ea+b = eaeb) and Euler’s formula. Here are some examples: (a + ib) + (c + id) = (a + c) + i(b + d) (a + ib)(c + id) = ac + (ib)c + a(id) + (ib)(id) = ac + i2bd + i(bc + ad) = (ac − bd) + i(bc + ad) c(a + ib) = (ca) + i(cb) a + ib (a + ib)(c − id) (ac + bd + i(bc − ad) = = c + id (c + id)(c − id) c2 + d2 ea+ib = eaeib = ea(cos b + i sin b) = ea cos b + iea sin b

1 Math 441, Summer 2018 A.J. Hildebrand

Complex Number Magic

• Computations with e, π, and i:

eiπ = cos π + i sin π = −1 eiπ/2 = cos(π/2) + i sin(π/2) = i πi = e(ln π)i = cos(ln π) + i sin(ln π)

• Power series with complex numbers and proof of Euler’s formula: Complex numbers can be substituted into power series as long as the series converges absolutely with z replaced by |z|. Here is a proof of Euler’s formula eiθ = cos θ + i sin θ using power series and the fact that i2 = −1, i3 = −i, i4 = 1, etc.:

(iθ) (iθ)2 (iθ)3 (iθ)4 eiθ = 1 + + + + + ··· 1! 2! 3! 4! iθ −θ2 −iθ3 θ4 = 1 + + + + + ··· 1! 2! 3! 4! θ2 θ4 θ θ3 = 1 − + − · · · + i − + ··· 2! 4! 1! 3! = cos θ + i sin θ.

• Trigonometric identities via complex numbers. Here is a derivation of the addition formulas for sin(a + b) and cos(a + b) using complex numbers:

ei(a+b) = eiaeib (by rules for exponentials) cos(a + b) + i sin(a + b) = (cos a + i sin a)(cos b + i sin b) (by Euler’s formula) cos(a + b) + i sin(a + b) = (cos a cos b − sin a sin b) + i(sin a cos b + cos a sin b) (multiply out) cos(a + b) = cos a cos b − sin a sin b (take real part on each side) sin(a + b) = sin a cos b + cos a sin b (take imaginary part on each side)