Simple Nature - Light and Matter

x / The complex number e iφlies on the unit circle.a result known as Euler’s formula. The geometrical interpretationin the complex plane is shown in figure x.Although the result may seem like something out of a freak showat first, applying the definition of the exponential function makes itclear how natural it is:e x = limn→∞(1 + x n) n.When x = iφ is imaginary, the quantity (1 + iφ/n) represents anumber lying just above 1 in the complex plane. For large n, (1 +iφ/n) becomes very close to the unit circle, and its argument is thesmall angle φ/n. Raising this number to the nth power multipliesits argument by n, giving a number with an argument of φ.Euler’s formula is used frequently in physics and engineering.Trig functions in terms of complex exponentials example 29⊲ Write the sine and cosine functions in terms of exponentials.⊲ Euler’s formula for x = −iφ gives cos φ−i sin φ, since cos(−θ) =cos θ, and sin(−θ) = − sin θ.cos x = eix + e −ix2sin x = eix − e −ix2iA hard integral made easy example 30⊲ Evaluate∫e x cos x dxy / Leonhard Euler (1707-1783)⊲ This seemingly impossible integral becomes easy if we rewritethe cosine in terms of exponentials:∫e x cos x dx∫ ( e= e x ix + e −ix )dx2= 1 ∫(e (1+i)x + e (1−i)x ) dx2= 1 ( )e(1+i)x+ e(1−i)x+ c2 1 + i 1 − iSince this result is the integral of a real-valued function, we’d likeit to be real, and in fact it is, since the first and second terms arecomplex conjugates of one another. If we wanted to, we coulduse Euler’s theorem to convert it back to a manifestly real result. 55 In general, the use of complex number techniques to do an integral couldresult in a complex number, but that complex number would be a constant,which could be subsumed within the usual constant of integration.606 Chapter 10 Fields