# Another HARD complex question

• Oct 22nd 2006, 08:43 AM
sg2004
Another HARD complex question
Need to do these by tomorrow morning. Any last minute help welcome. Cheers :)

Which of the following functions are holomorphic at z=0? Prove your answer
A) lzlsquared
B)Re(z) + Im(z)
C)Re(z).Im(z)
• Oct 22nd 2006, 10:56 AM
CaptainBlack
Quote:

Originally Posted by sg2004
Need to do these by tomorrow morning. Any last minute help welcome. Cheers :)

Which of the following functions are holomorphic at z=0? Prove your answer
A) lzlsquared
B)Re(z) + Im(z)
C)Re(z).Im(z)

(C) has been done in this recent thread, it is equivalent to problem (b) there.

RonL
• Oct 22nd 2006, 11:00 AM
CaptainBlack
Quote:

Originally Posted by sg2004
B)Re(z) + Im(z)

f(x+i y) = x+y

does not satisfy the Cauchy-Riemann equations at z=0.

(du/dx=1, but dv/dy=0)

RonL
• Oct 22nd 2006, 11:18 AM
CaptainBlack
Quote:

Originally Posted by sg2004
A) lzlsquared

Is not holomorphic at z=0, as though it satisfies the Cauchy-Riemann
equations there, it does not satisfy them at any other point, so it is
not complex differentiable in any neighbourhood of 0.

RonL
• Oct 23rd 2006, 12:48 AM
sg2004
Thanks so much Capn Black.... Cheers! :)