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1 Complex Numbers for Harmonic Functions |
Instead of using a complex conjugate pair to ensure a real number (Form 3 and Form 4), Form 3’ and Form 4’ use a brute force to take the real part of a complex number to get the real number.
The pros and cons of using Re[ ] functions (Form 3’ and Form 4’) are listed below:
Pros:
Form 3’ and Form 4’ do not require showing the whole complex conjugate pair. They only need to show one complex number, which is half of the complex conjugate pair. As a result, formulations using Form 3’ and Form 4’ are more compact than ones using Form 3 and Form 4.
Cons:
Re[ ] functions (Form 3’ and Form 4’) may mislead us into thinking that the real solution is just ignoring the imaginary part of the complex solution. To be mathematically correct, whenever Re[ ] functions are used, we should know that the real solution is the result of the addition of two complex solutions as in Form 3 and Form 4.
In conclusion, even though using Re[ ] functions (Form 3’ and Form 4’) might not be mathematically correct, Form 3’ and Form 4’ are widely used due to their compact formulation showing only half of a complex conjugate pair. Therefore, we need to know how to convert Form 3’ and Form 4’ back to Form 3 and Form 4. For this purpose, converting the Re[ ] functions to Form 3 and Form 4 is demonstrated in Example 1.2. Also, exercises to convert the Re[ ] functions to Form 3 and Form 4 are provided in the Homework Exercises section.
1.4Mathematical Identity
Four mathematical identities will be used in deriving the four equivalent forms for representing simple harmonic waves. Math 1 is Euler’s formula, and Math 2 will be explained and derived. Math 3 and Math 4 will be shown below for reference:
Mathematical Identity
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Math 2 |
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Math 3 |
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Math 4 |
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Math 1 is Euler’s formula. It defines a complex exponential function, e jθ, in polar coordinates using cosine and sine functions:
e jθ ¼ cos ðθÞ j sin ðθÞ |
ð1Þ |
which is shorthand for
1.4 Mathematical Identity |
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ejθ ¼ cos ðθÞ þ j sin ðθÞ |
ð2Þ |
e jθ ¼ cos ðθÞ j sin ðθÞ |
ð3Þ |
Math 2 provides a bridge between real numbers, cos(θ), and complex numbers, e jθ. The cosine function in Eq.(4) can be derived from Euler’s formulas by adding Eq.(2) and Eq.(3). The sine function in Eq.(5) can be derived from Euler’s formulas by subtracting Eq.(3) from Eq.(2):
cos ðθÞ ¼ |
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ejθ þ e jθ |
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sin ðθÞ ¼ |
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ejθ e jθ |
ð5Þ |
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Note that, in Eq.(4), both sides of the equation are real numbers. The left-hand side of the equation, cos(θ), is a real number. The right-hand side of the equation, 12 ejθ þ e jθ , is also a real number because the addition of a complex conjugate pair is a real number. Even though both sides of Eq.(4) are real, the formats are totally different. The left-hand side of the equation is formulated as a real trigonometric function, cos(θ). The right-hand side of the equation is formulated as complex exponential functions, 12 ejθ þ e jθ . Therefore, Math 2 is a bridge between real numbers and complex numbers.
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Math 3 is one of the trigonometric identities.
Math 4 is a property of exponential functions, valid for both real and complex a and b.