# Imaginary part

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

In terms of the complex conjugate [itex]\bar{z}[itex], the imaginary part of z is equal to [itex]\frac{z-\bar{z}}{2\mathrm{i}}[itex].

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

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