- #1
lazypast
- 85
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[tex] Re^{j \theta} = R{(cos \theta + jsin\theta )}[/tex]
can anyone show me this proof or show me a link please
can anyone show me this proof or show me a link please
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You can also prove this result using calculus. If you have met differentiation, try taking the derivative of f(x) as see where it takes you,lazypast said:I haven't come across this Taylor series much, using this I take it the proof is harder that i suspected
From a book i have read in complex analysis, and from what i understand about the subject, I believe that we just define this equation. I mean, we define that [tex] e^{a+j \theta} = e^a{(cos \theta + jsin\theta )}[/tex] . It's simply the definition for complex exponents. You can also reach this equation by using the Taylor series of e^x=1+x+(x^2)/2!+... , using x=ja but it's not any real proof unless you prove that the series converges for any imaginary number ja. Also, how can you check the above equation while the left part of it cannot be interpreted in a certain way?? So, i see it as a definition and not as an equation that you can prove.lazypast said:[tex] Re^{j \theta} = R{(cos \theta + jsin\theta )}[/tex]
can anyone show me this proof or show me a link please
lazypast said:I see now, silly mistakes by me.
When a derivative is zero then the things differentiated must have been a constant
lazypast said:well my original question has been answered now so thank you all.
as for that last this hootenanny do you mean e^ix= or f(x)=
f(x)=(1+0)/1=1 and e^ix: 1=1+0
but i don't understand what either of these prove
A complex number is a number that contains both a real and an imaginary part. It is typically written in the form a + bi, where a is the real part and bi is the imaginary part (with i representing the imaginary unit).
The "j" in Re^{jθ} represents the imaginary unit, which is equal to the square root of -1. It is used in complex numbers to distinguish the imaginary part from the real part.
The "θ" in Re^{jθ} represents the angle or phase of the complex number in polar form. It is measured in radians and determines the direction of the vector from the origin to the complex number.
To convert a complex number from rectangular form (a + bi) to polar form (Re^{jθ}), you can use the following formulas: R = √(a^2 + b^2) and θ = tan^-1(b/a). This will give you the magnitude and angle of the complex number.
The "e" in Re^{jθ} represents Euler's number, which is a mathematical constant equal to approximately 2.71828. It is used in complex numbers to convert from rectangular form to polar form, and vice versa.