1. PHY 301: MATH AND NUM TECH Contents Chapter 7: Complex Numbers I.Definitions and Representation II.Properties III.Differentiaton of complex functions

2. PHY 301: MATH AND NUM TECH I. Definitions and Representations

3. PHY 301: MATH AND NUM TECH I. Definitions and Representations B.Operations: Addition Multiplication: Zero element (w/ respect to +) Identity element (with respect to -) Inverse and division

4. PHY 301: MATH AND NUM TECH I. Definitions and Representations E7.1-2 Compute z -1 for z=3i ; z=1-2i; 1+2i. Write answer in the form z=a+ib where a and b are real. E7.1-3 Compute the magnitude of the following complex numbers by computing directly zz*: z=i z=1+i z= 1/(2-3i) z=(1+i)/(1-i)

5. PHY 301: MATH AND NUM TECH I. Definitions and Representations C.Argand (or complex) plane representation: E7.1-4 What transformation, in the complex plane, does the multiplication by i induce? (prove statement) Interpretation of multiplication z=z 1 z 2 as a transformation in Argand plane: E7.1-5 Find  and  for z=1+i; z=1; z=i; z=-2-3i, and z=-1 and write these numbers in angle magnitude (Argand variables) form: z=  (cos  +isin .

6. PHY 301: MATH AND NUM TECH I. Definitions and Representations E7.1-6 Write ; z=i, z=-1, z=-1-2i and z 1 =1+i, z 2 =1/(1+i) in exponential form then compute z 1 *z 2 and show that you get the expected results. General expression for z in exponential notation: Note how multiplication becomes easy:

7. PHY 301: MATH AND NUM TECH II. Fundamental Properties Roots of polynomials: Quadratic polynomials roots Roots of unity: z n =1 and nth root of a complex number E7.2-1 Find the cubic roots of unity and plot the solutions or the Argand plane

8. PHY 301: MATH AND NUM TECH III. Differentiation The derivative of f is well defined if the above limit exists and is unique. Notice that since is complex there are an infinite number of ways  can approach zero and thus we are lead to strict constraints on f to be differentiable. Let’s compute for a function f: However since  lives in the Argand plane there are only 2 independent ways to approach zero: along the real axis, or along the imaginary axis. Forcing these 2 paths to lead to the same limit imposes the so called Cauchy Riemann equations on f, leading us to a sufficient condition for f to be differentiable. First, for convenience we write: First let’s compute the limit when  approaches 0 along the real axis:  =  +i0 Then let’s compute the limit when  approaches 0 along the imaginary axis:  = 0+i 

9. PHY 301: MATH AND NUM TECH III. Differentiation In order for the derivative to be meaningful we thus have to have these 2 limits equal. This will guaranty that no matter how z approaches zero (ie along what direction in the complex plane) the limit will be unique and thus the derivative will be well defined and unique. Equating the real parts of the previous equations as well as the imaginary parts, we get the following set of 2 conditions, called Cauchy-Riemann equations, for a function to be differentiable with respect to the complex variable z=x+iy:

10. PHY 301: MATH AND NUM TECH III. Differentiation