University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2022
Views:

242

Pages:

1
Author:

Arabella Ferguson

Summary Sheet: Complex Numbers Definition: A complex number is anything of the form a + bi, where a, b are real and √ i = −1. • The real numbers are√a subset of the complex numbers (set b = 0 to get the real numbers). Since i = −1, we can easily compute the powers of i: i2 = −1, i3 = −i, i4 = 1, etc. • The conjugate of a + bi is a − bi. Notation: a + bi = a − bi • The “real part” of a+bi is a, the imaginary part of a+bi is b. Notation: Re(a+bi) = a, Im(a + bi) = b. q √ • The modulus, or size, of a complex number: |a + bi| = (a + bi)(a − bi) = a2 + b2 . • Complex numbers can be plotted in two dimensions. The complex number a + bi is plotted as (a, b). Notice that |a + bi| is the distance from (a, b) to (0, 0). • Euler’s Formula: eiθ = cos(θ) + i sin(θ). From this, we get one of the most elegant equations of mathematics: eiπ + 1 = 0 Note that this gives all of the big constants of mathematics in a single beautiful relationship. Complex Arithmetic is very similar to polynomial arithmetic: 1. Addition/subtraction: (a + bi) ± (c + di) = (a ± c) + (b ± d)i 2. Multiplication: (a + bi)(c + di) = ac + adi + bci + bdi2 = (ac − bd) + (ad + bc)i 3. Division: Multiply by the conjugatea + bi a + bi c − di (a + bi)(c − di) = · = c + di c + di c − di c2 + d2 The Polar Form of a Complex Number We already know the polar form of an√ordered pair, (a, b). In calculus, we learned that a = R cos(θ), b = R sin(θ), where R = a2 + b2 and θ is the (four-quadrant) angle that a line from (0, 0) to (a, b) would make with the positive x−axis. Therefore, a + bi ⇒ (a, b) ⇒ (R cos(θ), R sin(θ)) ⇒ R(cos(θ) + i sin(θ)) We can therefore write a + bi as R(cos(θ) + i sin(θ). Using Euler’s formula, we can write a + bi simply as Reiθ . Using this, we can define powers and logs using i: • ln(a + bi) = ln(Reiθ ) = ln(R) + iθ • aβi = ei ln(a) We have extended logarithms to negative real numbers- Note that ln(−1) = ln(1) + i · π, so ln(−1) = πi. You can also do things like: ii = e−π/2 Rules of Exponents Still Apply For example, ea+bi = ea ebi , etc. 1

Summary Sheet: Complex Numbers - Study Guide

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