3—Complex Algebra 78
3.21 What isii? Ans: If you’d like to check your result, typei∧iintoGoogle.
3.22 For what argument doessinθ= 2?cosθ? Ans: 1. 5708 ±i 1. 3170
3.23 What are the other trigonometric functions,tan(ix),sec(ix), etc. What aretanandsecfor the general
argumentx+iy. Ans:tan(x+iy) = (tanx+itanhy)/(1−itanxtanhy)
3.24 The diffraction pattern from a grating involves the sum of waves from a large number of parallel slits. For
light observed at an angleθaway from directly ahead, this sum is, forN+ 1slits,
d
d
d
d
d
dsinθ
θ
r 0
r 0 −dsinθ
cos
(
kr 0 −ωt
)
+ cos
(
k(r 0 −dsinθ)−ωt
)
+ cos
(
k(r 0 − 2 dsinθ)−ωt
)
+
...+ cos
(
k(r 0 −Ndsinθ)−ωt
)
Express this as the real part of complex exponentials and sum the finite series. Show that the resulting wave is
sin
( 1
2 (N+ 1)kdsinθ
)
sin
( 1
2 kdsinθ
) cos
(
k(r 0 −^12 Ndsinθ)−ωt
)
Interpret this result as a wave that appears to be coming from some particular point (where?) and with an
intensity pattern that varies strongly withθ.