A complex function ๐‘“(๐‘ง) = ๐‘ข(๐‘ฅ, ๐‘ฆ) + ๐‘– ๐‘ฃ(๐‘ฅ, ๐‘ฆ) and its complex conjugate ๐‘“โˆ—(๐‘ง) = ๐‘ข(๐‘ฅ, ๐‘ฆ) โˆ’ ๐‘– ๐‘ฃ(๐‘ฅ, ๐‘ฆ) are both analytic in the entire complex plane

Q. A complex function ๐‘“(๐‘ง) = ๐‘ข(๐‘ฅ, ๐‘ฆ) + ๐‘– ๐‘ฃ(๐‘ฅ, ๐‘ฆ) and its complex conjugate ๐‘“โˆ—(๐‘ง) = ๐‘ข(๐‘ฅ, ๐‘ฆ) โˆ’ ๐‘– ๐‘ฃ(๐‘ฅ, ๐‘ฆ) are both analytic in the entire complex plane, where ๐‘ง = ๐‘ฅ + ๐‘– ๐‘ฆ and ๐‘– = โˆšโˆ’1 . The function ๐‘“ is then given by

Sol:

f(z) = u(x, y) + i v(x, y)

f*(z) = u (x, y) – i v(x, y)

f(z) is analytic function

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