Tricks with Complex Number phys3330 – Spring 2012 Things you need to know about the complex number for this course: 1.Perform algebraic operations on complex number and represent a given complex number graphically and express it in polar form. 2. Represent a sinusoidal function as the real and imaginary part of an exponential and use this representation for adding trigonometric functions. 3. Set up a linear differential equation to describe the behavior of LCR circuit and subject to a sinusoidal applied voltage. 4. Use complex exponentials to solve homogenous and inhomogeneous linear differential equation with constant coefficient.

1. Complex number - The imaginary unit: and (a)General complex number (where x and y are REAL number) real part of z, x= Re z imaginary part of z, x= Im z (b) Arithmetic in complex number : just like ordinary + − × ÷ (c) Complex conjugate (z → z*) : Replacing i to -i → ✱ ✱ ✱ (used in rationalize the denominator) ✱ → the modulus of z

2. Power Series for exponential and trigonometric functions: Now compare trigonometric and hyperbolic function in complex number:

3. Polar representation of a complex number z=x+iy x (real) z=x+iy r ϕ y (imaginary) Representing z=x+iy by the point (x,y) Then, where, Now we can always write x y ✵Polar representation is advantageous for multiplication and division! Let then ✵Example. Find z satisfying z3 = -8i (solution) Since they all have modulus 2. Thus Now, thus, and, So,

where A(t) is a slowly varying function of time. Homework Problems 1. Let (a) Represent z1 and z2 in the complex plane and find their real and imaginary part (a) Evaluate z1 + z2 and 2. By writing out cos θ in terms of exponentials and using the binomial expansion, express (cos θ)5 in terms of cos θ, cos 3θ and cos 5θ. 3. Evaluate the sum 4. Suppose that frequencies and differ only slightly. Use the complex exponential, express the sum (A is a constant) in the form of where A(t) is a slowly varying function of time.