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>.
Sometimes computations with real periodic functions such as alternating currents and electromagnetic fields are simplified by writing them as the real parts of complex functions. See for example electrical impedance.
[edit] See also
eo:Vikipedio:Projekto matematiko/Reela parto it:Parte reale pt:Parte real
