Chem 263 Math Review 1.1 Review of Complex Exponential Functions. A sinusoidal function can be written as an exponential function with a complex exponent: where (i is called the imaginary unit) and is the second equality, i.e.: Euler’s Formula Here, is a real number given in radians (). Notice that both the real and imaginary values of oscillate 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 allow one to solve equations that have no real -1 solution. For example, the solution to the equation is Here, realize that -i 1 Complex numbers can also be written in polar coordinates as , where r is the and the The magnitude is . Clearly the magnitude of is unity (i.e., = 1) magnitude phase. for all values of the phase. The plot of is a cosine wave, while that of is a sine wave. The two waves are out of phase with each π/2 other: 1.2 Plane Waves.
Chem 263 Math Review 1.1 Review of Complex Exponential Functions. A sinusoidal function can be written as an exponential function with a complex exponent: where (i is called the imaginary unit) and Euler’s Formula is the second equality, i.e.: Here, is a real number given in radians (). Notice that both the real and imaginary values of oscillate 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 allow one to solve equations that have no real solution. For example, the solution to the equation is Here, realize that 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 (i.e., = 1) for all values of the phase. The plot of is a cosine wave, while that of is a sine wave. The two waves are π/2 out of phase with each other: 1.2 Plane Waves.
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