If #kw^2+kww"* "+iw-iw"*"=0#, where #w# is a complex number and #w"*"# is its conjugate, then show that #w# is either pure real number or pure imaginary number?

1 Answer
Jul 23, 2017

Please see below.

Explanation:

Let #w=a+bi# then #w"*"=a-bi# and then

#kw^2+kww"* "+iw-iw"*"=0# changes to

#k(a+bi)^2+k(a+bi)(a-bi)+i(a+bi)-i(a-bi)=0+i0#

or #k(a^2-b^2+2iab)+k(a^2+b^2)+ia-b-ia-b=0+i0#

or #2ka^2-2b+2iabk=0+i0#

i.e. #2ka^2-2b=0# or #b=a^2k#

and as #2abk=0# and #k# is non-zero,

either #b=0# or #a=0#

i.e. either #w# is purely imaginary or purely real.