Complex Numbers and Series

Here are the central concepts and results in our unit on complex numbers and series, which can be found on the webpage with url

 http://www.math.umd.edu/undergraduate/courses/dcp/141/141cxnotes.pdf

Definition 1.1: A complex number is a number z of the form z = x + iy (or equivalently, z = x + yi), where x and y are real numbers, and where i^{2} = −1.

Comment: The set of all complex numbers is denoted C and the set of real numbers (that is, all z such that z = x) is denoted R.

Comment: Each complex number a + ib can be considered as a point in the plane, with a representing the x coordinate and b the y coordinate.

Addition of complex numbers: (a + ib) + (c + id) = (a + c) + i(b + d) Multiplication of complex numbers: (a + ib)(c + id) = (ac − bd) + i(ad + bc)

Distance between z = a + ib and 0: |z| = \sqrt{a^{2} + b^{2}}. The number |z| is the modulus of z.

Distance between z_{1} = a + ib and z_{2} = c + id: |z_{1} − z_{2}| = \sqrt{(c − a)^{2} + (d − b)^{2}}

Theorem 5.1 – Fundamental Theorem of Algebra: Every nonconstant polynomial with coefficients in C (or R) has a root in C.

Theorem 5.2 – Factorization Theorem: Suppose p is a polynomial of degree n ≥ 1 and with complex coefficients, so p(z) = c_{n}z^{n} +c_{n−1}z^{n−1} +⋯+c_{1}z +c_{0}, with c_{n} ≠ 0. Then p can be factored as a product of linear terms

p(z) = c_{n}(z − z_{1})(z − z_{2}) ⋯ (z − z_{n})

where the numbers z_{1}, z_{2}, …, z_{n} are the roots of p. (Possibly some roots appear more than once.)

Derivative of f: If the limit exists, then f ′(z_{0}) = lim_{z→z_{0}} \frac{f(z) − f(z_{0})}{z − z_{0}}.

Comment: Derivatives for complex polynomials and many other complex functions are obtained just like the corresponding ones for real functions. Thus if f(z) = z^{5}, then df/dz = 5z^{4}.

Infinite series: A complex infinite series has the form ∑_{n=1}^{∞} b_{n}, where the terms b_{1}, b_{2}, … are complex numbers. The series converges to the complex number L provided that the partial sums s_{n} converge to L in the complex plane, that is, if

lim_{n→∞}(b_{1} + b_{2} + ⋯ + b_{n}) = lim_{n→∞} s_{n} = L

Power series: A complex power series has the form ∑_{n=0}^{∞} a_{n}z^{n}, where the terms a_{n} are complex numbers and z is a (complex) variable.

Power series for exponential, sine, cosine functions:

e^{z} = 1 + z + \frac{z^{2}}{2!} + \frac{z^{3}}{3!} + ⋯

sin z = z − \frac{z^{3}}{3!} + \frac{z^{5}}{5!} − + ⋯

cos z = 1 − \frac{z^{2}}{2!} + \frac{z^{4}}{4!} − + ⋯

Euler’s Formula: For any complex number z, e^{iz} = cos z + i sin z Polar form of z: z = x + iy = R cos θ + iR sin θ = Re^{iθ} Products in polar form: If z_{1} = R_{1}e^{iθ_{1}} and z_{2} = R_{2}e^{iθ_{2}}, then

z_{1}z_{2} = R_{1}e^{iθ_{1}}R_{2}e^{iθ_{2}} = R_{1}R_{2}e^{iθ_{1}}e^{iθ_{2}} = (R_{1}R_{2})e^{i(θ_{1}+θ_{2})}

Thus for products, moduli are multiplied and angles are added.

Non-WebAssign Exercises:

1. Plot in the complex plane the following:

(a) The points i, 2 − i, 2 + i, −5

(b) The collection of points z such that |z| = 1/2

(c) The collection of points z such that |z − 1 + 3i| = 2

(d) The points z, w, and zw, where z = 3e^{iπ/4} and w = 2e^{4iπ/3}

2. Find all (complex) solutions of the given equation.

(a) 4z^{2} + 27 = 0

(b) z^{2} − z + 1 = 0

(c) z^{2} + 3z + 3 = 0

3. Let z = 1 − i and w = \sqrt{3} − i. Find the polar form of zw and z/w, and plot them in the complex plane.

4. Let f(z) = −4z^{5} + 3z^{2} − 2z + 1/z. Find the derivative of the function f.

5. Write out the first 5 terms of (a) \{i^{n}\}_{n=1}^{∞} (b) \{(\frac{1+i}{2})^{n}\}_{n=1}^{∞}

6. Like for real power series, a complex power series ∑_{n=0}^{∞} c_{n}z^{n} either converges for all complex numbers z or converges only for z = 0, or else there is a positive (real) number R such that ∑_{n=0}^{∞} c_{n}z^{n} converges for |z| < R and diverges for |z| > R. We write R = ∞ if the power series converges for all z, and R = 0 if the power series converges only for z = 0. We call R the radius of convergence of the power series. Find the radius of convergence R for

(a) e^{z} = ∑_{n=0}^{∞} \frac{z^{n}}{n!}  (b) ∑_{n=0}^{∞} z^{n}  (c) ∑_{n=0}^{∞}(1 + i)^{n}z^{n}  (d) ∑_{n=0}^{∞} 3^{n}z^{2n}