Imaginary part

In mathematics, the imaginary part of a complex number z, is the second element of the ordered pair of real numbers representing z, i.e. if z = (x, y) , or equivalently, z = x+iy, then the imaginary part of z is y. It is denoted by mbox{Im}z or Im z. The complex function which maps z to the imaginary part of z is not holomorphic.

Related Topics:
Mathematics - Complex number - Real number - Complex function - Holomorphic

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In terms of the complex conjugate ar{z}, the imaginary part of z is equal to rac{z-ar{z}}{2i}.

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For a complex number in polar form, z = (r, heta ), or equivalently, z = r(cos heta + i sin heta) , it follows from Euler's formula that z = re^{i heta}, and hence that the imaginary part of re^{i heta} is rsin heta.

Related Topics:
Polar form - Euler's formula

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In electric power, when a sine wave voltage drives a "linear" load (in other words, a load that makes the current also be a sine wave),

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the current I in the power wires can be represented as a complex number I = x + jy.

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(Engineers use "j" to indicate the imaginary unit rather than "i". To them, "i" represents current).

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The "real current" x is related to the current when the voltage is maximum.

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The real current times the voltage gives the actual power consumed by the load

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(often all that power is dissipated as heat).

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The "imaginary current" y is related to the current when the voltage is zero.

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A load with purely imaginary current

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(such as a capacitor or inductor)

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dissipates no power;

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it merely accepts power temporarily then later pushes that power back on the power lines.

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