A complex function ๐(๐ง) = ๐ข(๐ฅ, ๐ฆ) + ๐ ๐ฃ(๐ฅ, ๐ฆ) and its complex conjugate ๐โ(๐ง) = ๐ข(๐ฅ, ๐ฆ) โ ๐ ๐ฃ(๐ฅ, ๐ฆ) are both analytic in the entire complex plane
Q. A complex function ๐(๐ง) = ๐ข(๐ฅ, ๐ฆ) + ๐ ๐ฃ(๐ฅ, ๐ฆ) and its complex conjugate ๐โ(๐ง) = ๐ข(๐ฅ, ๐ฆ) โ ๐ ๐ฃ(๐ฅ, ๐ฆ) are both analytic in the entire complex plane, where ๐ง = ๐ฅ + ๐ ๐ฆ and ๐ = โโ1 . The function ๐ is then given by
Sol:
f(z) = u(x, y) + i v(x, y)
f*(z) = u (x, y) โ i v(x, y)
f(z) is analytic function