Imaginary part
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In mathematics, the imaginary part of a complex number <math> z<math>, is the second element of the ordered pair of real numbers representing <math>z,<math> i.e. if <math> z = (x, y) <math>, or equivalently, <math>z = x+\mathrm{i}y<math>, then the imaginary part of <math>z<math> is <math>y<math>. It is denoted by <math>\mbox{Im}z<math> or <math>\Im z<math>. The complex function which maps <math> z<math> to the imaginary part of <math>z<math> is not holomorphic.
In terms of the complex conjugate <math>\bar{z}<math>, the imaginary part of z is equal to <math>\frac{z-\bar{z}}{2\mathrm{i}}<math>.
For a complex number in polar form, <math> z = (r, \theta )<math>, or equivalently, <math> z = r(cos \theta + \mathrm{i} sin \theta) <math>, it follows from Euler's formula that <math>z = re^{\mathrm{i}\theta}<math>, and hence that the imaginary part of <math>re^{\mathrm{i}\theta} <math> is <math>r\sin\theta<math>.