I have these functions below: $$\sqrt{(x+iy)^2-a^2}$$
$$\frac{b(x+iy)}{\sqrt{(x+iy)^2-a^2}}$$
How do I split these to get the real and imaginary parts of these functions?
If anyone could help me out, that really would be helpful!!!!
It will help me with the method of manufactured solutions for my fracture problem that has the Westergaard solution (as the analytical solution)!!!
Thank you very much,
Mousumi
Best Answer
Let $\sqrt{(x+iy)^2-a^2}=c+id$
$\implies (x+iy)^2-a^2=(c+id)^2$
Comparing the real & the imaginary parts
$\implies c^2-d^2=x^2-y^2-a^2\ \ \ \ (1)$
$2cd=2xy$
$(c^2+d^2)^2=(c^2-d^2)^2+(2cd)^2=(x^2-y^2-a^2)^2+(2xy)^2=(x^2+y^2)^2+a^4-2a^2(x^2-y^2)$
$c^2+d^2=\sqrt{(x^2+y^2)^2+a^4-2a^2(x^2-y^2)}\ \ \ \ (2)$
Use $(1),(2)$
Observe that $cd$ will have the same sign as $xy$