Real part
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In mathematics, the real part of a complex number <math> z<math>, is the first 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+iy<math>, then the real part of <math>z<math> is <math>x<math>. It is denoted by <math>\mbox{Re}z<math> or <math>\Re z<math>. The complex function which maps <math> z<math> to the real part of <math>z<math> is not holomorphic.
In terms of the complex conjugate<math>\bar{z}<math>, the real part of <math>z<math> is equal to <math>z+\bar z\over2<math>.
For a complex number in polar form, <math> z = (r, \theta )<math>, or equivalently, <math> z = r(cos \theta + i \sin \theta) <math>, it follows from Euler's formula that <math>z = re^{i\theta}<math>, and hence that the real part of <math>re^{i\theta} <math> is <math>r\cos\theta<math>.