What does sin(z with a hat) mean?

Question

Sorry for the non-technical term, but I need to use cauchy-riemann equations on sin(z hat) and cos(z hat) to prove they're not analytic... which I'm sure I could try if I knew what it mean... I know for elementary maths that it means conjugate... but I can't find anywhere in my text book which one to conj? If thats even what it is...

This question is in my text book after the section that talks about how

sin(z) = [e^(iz) - e^(-iz) ] /2i

so if thats the equation they want me to d/dx then what is it's "hat" :) ....

Answer 1

This is complex variables and the hat comes from engineering to signify it’s a vector depending on the discipline it could be a vector or a unit vector. In this case it is a vector defined as:

z hat = x + i y
f = sin(z hat) = sin(x + i y) = sin x cos iy + cos x sin iy
from the second link below sin(x+iy)=sin(x)cosh(y)
+i.cos(x)sinh(y).

so
f = u(x,y) + v(x,y)
u(x,y) = sin(x)cosh(y)
and
v(x,y) = icos(x)sinh(y)

and from the first link below:

partial u/ partial x = cos(x)cosh(y)
partial v/ partial y = icos(x)cosh(y)

and
partial v/ partial x = -isin(x)sinh(y)
partial u/ partial y = sin(x)sinh(y)

partial u/ partial x != partial v/ partial y
ie.
cos(x)cosh(y) != icos(x)cosh(y)
so the function is not analytic


If I had more time I would have proved
sin(x+iy)=sin(x)cosh(y)
+i.cos(x)sinh(y)
but it is straight forward.

I hope this helps.

Answer 2

This is not a notation I have ever seen with complex numbers. However, I have seen i hat and j hat used to mean unit vectors i and j. This is pretty unnecessary though, as i and j are normally unit vectors anyway.

I would guess that z hat is a complex number of unit modulus, so that the real part is cos z and the imaginary part is i sin z.