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**Example text**

18) is called the polar form of z. It is immediately clear that, the complex conjugate of z in the polar form is z ∗ (r, θ) = z (r, −θ) = re−iθ . In the complex plane, z ∗ is the reﬂection of z across the x-axis. It is helpful to always keep the complex plane in mind. As θ increases, eiθ describes an unit circle in the complex plane as shown in Fig. 3. To reach a general complex number z, we must take the unit vector eiθ that points at z and stretch it by the length |z| = r. It is very convenient to multiply or divide two complex numbers in polar forms.

For example, we can interpret z 1/4 as the fourth root of z. In other words, we want to ﬁnd a number whose 4th power is equal to z. It is instructive to work out the details for the case of z = 1. Clearly 14 = ei0 4 = ei0 = 1, 4 i4 = eiπ/2 4 (−1) = eiπ 4 4 = ei2π = 1, = ei4π = 1, (−i) = ei3π/2 4 = ei6π = 1. Therefore there are four distinct answers ⎧ ⎪ ⎪ 1 , ⎨ i , 1/4 1 = −1 , ⎪ ⎪ ⎩ −i . The multiplicity of roots is tied to the multiple ways of representing 1 in the polar form: ei0 , ei2π , ei4π , etc.

I θ− θ3 θ5 + + ··· 3! 5! Now it was already known in Euler’s time that the two series appearing in the parentheses are the power series of the trigonometric functions cos θ and sin θ, respectively. 2) eiθ = cos θ + i sin θ, which at once links the exponential function to ordinary trigonometry. Strictly speaking, Euler played the inﬁnite series rather loosely. Collecting all the real terms separately from the imaginary terms, he changed the order of terms. To do so with an inﬁnite series can be dangerous.