Math Problem Statement

express the right hand side of f(z) =x^2 -y^2 -2y +i^2x + i^2xy in terms of z

Solution

We want to express the right-hand side of the function f(z)=x2y22y+i2x+i2xyf(z) = x^2 - y^2 - 2y + i^2x + i^2xy in terms of zz, where z=x+iyz = x + iy, with xx and yy being the real and imaginary parts of zz, respectively.

Step 1: Simplify the terms involving i2i^2

Recall that i2=1i^2 = -1. This allows us to rewrite the terms involving i2i^2:

i2x=xandi2xy=xyi^2x = -x \quad \text{and} \quad i^2xy = -xy

Thus, the expression becomes:

x2y22y+(x)+(xy)x^2 - y^2 - 2y + (-x) + (-xy)

Simplifying further:

x2y22yxxyx^2 - y^2 - 2y - x - xy

Step 2: Express in terms of z=x+iyz = x + iy

We now aim to rewrite this expression in terms of zz. Using the identity z=x+iyz = x + iy and its conjugate zˉ=xiy\bar{z} = x - iy, we know the following relations:

  • x=z+zˉ2x = \frac{z + \bar{z}}{2}
  • y=zzˉ2iy = \frac{z - \bar{z}}{2i}

Using these, substitute xx and yy into the expression x2y22yxxyx^2 - y^2 - 2y - x - xy.

Step 3: Substitution and expansion

Let me work through the substitution to obtain the expression in terms of zz.The right-hand side of the expression, after substitution and simplification, in terms of zz and zˉ\bar{z} (the conjugate of zz), is:

f(z)=(2+i)4z2+(1+2i)2z+(2i)4zˉ2(1+2i)2zˉf(z) = \frac{(2 + i)}{4}z^2 + \frac{(-1 + 2i)}{2}z + \frac{(2 - i)}{4}\bar{z}^2 - \frac{(1 + 2i)}{2}\bar{z}

This expression is now fully represented in terms of zz and zˉ\bar{z}, with no explicit xx or yy.

Would you like a detailed breakdown of how each term was substituted?

Here are 5 related questions for further exploration:

  1. How would the expression change if the function was f(z)=x2y2+i2f(z) = x^2 - y^2 + i^2?
  2. What is the geometric interpretation of functions involving both zz and zˉ\bar{z}?
  3. Can we express this function entirely in terms of zz without involving zˉ\bar{z}?
  4. How do complex conjugates affect the symmetry of a function in the complex plane?
  5. What are the conditions for a function of zz to be holomorphic?

Tip: Using complex conjugates often simplifies expressions in complex analysis, especially when converting between real and imaginary parts.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Algebra
Complex Conjugates

Formulas

z = x + iy
i^2 = -1
x = (z + \bar{z}) / 2
y = (z - \bar{z}) / 2i

Theorems

Complex Number Operations
Conjugation Properties

Suitable Grade Level

Undergraduate Level (Complex Analysis)