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# ComplexNumbersSHO - previous index next From a Circling...

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previous index next From a Circling Complex Number to the Simple Harmonic Oscillator Michael Fowler Describing Real Circling Motion in a Complex Way We’ve seen that any complex number can be written in the form i z re θ = , where r is the distance from the origin, and θ is the angle between a line from the origin to z and the x -axis. This means that if we have a set of numbers all with the same r , but different θ ’s, such as , i i re re α β , etc., these are just different points on the circle with radius r centered at the origin in the complex plane. Now think about a complex number that moves as time goes on: ( ) . i t z t Ae ω = At time t , z ( t ) is at a point on the circle of radius A at angle t ω to the x -axis. That is, z ( t ) is going around the circle at a steady angular velocity ω . We can also write this: ( ) cos sin i t z t Ae A t iA ω t ω ω = = + and see that the point z = x + iy is at coordinates ( ) ( ) , cos , sin . x y A t A t ω ω = x y O ( ) z t t ω sin A t ω A cos A t ω The angular velocity is ω

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ComplexNumbersSHO - previous index next From a Circling...

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