38 AN ELEMENTARY THEORY OF PHASE MICROSCOPY 



By definition 



e"^ = cos + i sin d) 

 e~'^ = cos — i sin 0. 



Hence any complex number can be written in the form 



z = a (cos d -\- i sin 6) 



ae 



(4.4) 

 (4.5) 



(4.6) 



in which a is the amphtude and d is the argument or phase of the complex 

 number. Equation 4.6 states the two forms of complex numbers as we 



a sin 6 



Fig. 11.10. Representation of a complex number 

 z = X + iy = o(cos 9 + i sin d) as a vector OP in the XY plane. 



shall use them in the theory of phase microscopy. It is worth noting 

 that 



z = a when 6 = 0; 



z = —a when = tt; (4.7) 



z = ±ia according as = db7r/2. 



Suppose that we are given two complex numbers z in the form 



zi = ai(cos0i + ?'sin0i); 



■22 = a2(cos 02 + i sin 62). 



The rule of addition is 



zi -\- Z2 = (ai cos ^1 + a2 cos ^2) + ^'(c^i sin di + 02 sin 62). (4.8) 



To add two complex numbers, add their real parts and their imaginary 

 parts. 



In order to multiply two complex numbers, it is more convenient and 



