Mathematical Tools for Physics

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.