Consider a complex functionf(x. y) = eax + i In by.If the function is analytic at (0. 1) then the possible values of (a,b) isa)( -1, 1)b)(1,- 1)c)( - 1, - 1)d)( 1, 1 )Correct answer is option 'D'. Can you explain this answer?

Question

Answer

Complex Function:
The given complex function is f(x, y) = e^(ax) * i * sin(by), where a and b are constants.

Analytic Function:
A complex function f(z) is said to be analytic at a point z = z0 if it is differentiable at z0 and also differentiable in a neighborhood of z0. In other words, the function must satisfy the Cauchy-Riemann equations at that point.

Cauchy-Riemann Equations:
The Cauchy-Riemann equations for a complex function f(z) = u(x, y) + iv(x, y) are:
∂u/∂x = ∂v/∂y (1)
∂u/∂y = -∂v/∂x (2)

Applying Cauchy-Riemann Equations:
Let's apply the Cauchy-Riemann equations to the given function f(x, y) = e^(ax) * i * sin(by).

∂u/∂x = ∂/∂x (e^(ax) * i * sin(by)) = a * e^(ax) * i * sin(by)
∂v/∂y = ∂/∂y (e^(ax) * i * sin(by)) = b * e^(ax) * i * cos(by)

∂u/∂x = ∂v/∂y
a * e^(ax) * i * sin(by) = b * e^(ax) * i * cos(by)

∂u/∂y = -∂v/∂x
∂/∂y (e^(ax) * i * sin(by)) = -∂/∂x (e^(ax) * i * sin(by))
a * e^(ax) * i * cos(by) = -b * e^(ax) * i * sin(by)

Simplifying the Equations:
From the first equation, we have:
a * e^(ax) * i * sin(by) = b * e^(ax) * i * cos(by)

Dividing both sides by e^(ax) * i, we get:
a * sin(by) = b * cos(by)

From the second equation, we have:
a * e^(ax) * i * cos(by) = -b * e^(ax) * i * sin(by)

Dividing both sides by e^(ax) * i, we get:
a * cos(by) = -b * sin(by)

Solving the Equations:
From the equation a * sin(by) = b * cos(by), we can divide both sides by sin(by) (assuming sin(by) is not equal to 0), and we get:
a = b * cot(by)

From the equation a * cos(by) = -b * sin(by), we can divide both sides by cos(by) (assuming cos(by) is not equal to 0), and we get:
a = -b * tan(by)

Equating the two values of a, we have:
b * cot(by) = -b * tan(by)

Dividing both sides by b and simplifying, we get:
cot(by) = -tan(by)

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