1.1 Review of Complex Exponentials. A sinusoidal function can be written as an exponential function with a complex exponent: where and is the second equality: Euler’s Formula Here, is a real number given in radians (). Notice that both the real and complex value of oscillates with a period of The function itself oscillates between pure real and pure imaginary values. We can tabulate the values and plot the function on the complex plane: A complex number can be written in the form of , where and are real x y numbers. The is and the is real part imaginary part x y. 0 1 i Complex numbers can also be written in polar coordinates as , where r -1 is the and the The magnitude is . Clearly the magnitude phase. -i magnitude of is unity for all values of the phase. 1 The plot of is a cosine wave, while that of is a sine wave: 1.2 Plane Waves. use powerpoint
1.1 Review of Complex Exponentials. A sinusoidal function can be written as an exponential function with a complex exponent: where and Euler’s Formula is the second equality: Here, is a real number given in radians (). Notice that both the real and complex value of oscillates with a period of The function itself oscillates between pure real and pure imaginary values. We can tabulate the values and plot the function on the complex plane: 0 1 i -1 -i 1 A complex number can be written in the form of , where x and y are real numbers. The real part is x and the imaginary part is y. Complex numbers can also be written in polar coordinates as , where r is the magnitude and the phase. The magnitude is . Clearly the magnitude of is unity for all values of the phase. The plot of is a cosine wave, while that of is a sine wave: 1.2 Plane Waves. use powerpoint
Reciprocal%20Lattice.docx