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Math  /  Calculus

Question8. Find constants a,ba, b so that f(z)=−x2+2x+y2+i(ay+bxy)f(z)=-x^{2}+2 x+y^{2}+i(a y+b x y) is analytic. A. a=2,b=0a=2, b=0 (B) a=2,b=−2a=2, b=-2 C. a=−2,b=2a=-2, b=2 D. a=−2,b=0a=-2, b=0

Studdy Solution

STEP 1

What is this asking? We need to find the right values of aa and bb that make the complex function f(z)f(z) analytic! Watch out! Remember, for a complex function to be analytic, it needs to satisfy the Cauchy-Riemann equations.
Don't forget those crucial partial derivatives!

STEP 2

1. Define the function
2. Apply Cauchy-Riemann Equations
3. Solve for *a* and *b*

STEP 3

Alright, let's break down our complex function f(z)f(z) into its real and imaginary parts.
We can write f(z)=u(x,y)+iv(x,y)f(z) = u(x, y) + iv(x, y), where u(x,y)u(x, y) is the real part and v(x,y)v(x, y) is the imaginary part.

STEP 4

Looking at our function f(z)=−x2+2x+y2+i(ay+bxy)f(z) = -x^2 + 2x + y^2 + i(ay + bxy), we can see that: u(x,y)=−x2+2x+y2u(x, y) = -x^2 + 2x + y^2 and v(x,y)=ay+bxyv(x, y) = ay + bxy Great! We've **identified** our real and imaginary components.

STEP 5

Now for the Cauchy-Riemann equations!
These are the key to analyticity.
They tell us that if a complex function is analytic, then its partial derivatives must satisfy these special relationships: ∂u∂x=∂v∂y\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and ∂u∂y=−∂v∂x\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}

STEP 6

Let's **calculate** those partial derivatives! ∂u∂x=∂∂x(−x2+2x+y2)=−2x+2\frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(-x^2 + 2x + y^2) = -2x + 2 ∂u∂y=∂∂y(−x2+2x+y2)=2y\frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(-x^2 + 2x + y^2) = 2y∂v∂x=∂∂x(ay+bxy)=by\frac{\partial v}{\partial x} = \frac{\partial}{\partial x}(ay + bxy) = by∂v∂y=∂∂y(ay+bxy)=a+bx\frac{\partial v}{\partial y} = \frac{\partial}{\partial y}(ay + bxy) = a + bx

STEP 7

Now, let's **plug these partial derivatives** into the Cauchy-Riemann equations: −2x+2=a+bx-2x + 2 = a + bx 2y=−by2y = -by

STEP 8

Looking at the first equation, −2x+2=a+bx-2x + 2 = a + bx, we can **compare the coefficients** of xx and the constant terms on both sides.
This gives us two equations: −2=b-2 = b and 2=a2 = a

STEP 9

The second equation, 2y=−by2y = -by, tells us that 2=−b2 = -b.
Since we already found that b=−2b = -2 from the first equation, this confirms our result!

STEP 10

So, we have a=2a = \mathbf{2} and b=−2b = \mathbf{-2}.

STEP 11

The constants that make f(z)f(z) analytic are a=2a = 2 and b=−2b = -2, which corresponds to option (B).

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