Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

22.) evaluate the cross ratio (2,1+i,1,1+i)(2, 1 + i, 1, 1 + i) b) identify the points where the mapping f(z)=12∗(z+1z)f(z) = \frac{1}{2} * (z + \frac{1}{z}) is conformal.

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Grade 8right chevron icon
Divide numbers written in scientific notationright chevron icon

Full solution

Q. 22.) evaluate the cross ratio (2,1+i,1,1+i)(2, 1 + i, 1, 1 + i) b) identify the points where the mapping f(z)=12∗(z+1z)f(z) = \frac{1}{2} * (z + \frac{1}{z}) is conformal.
  1. Calculate Cross Ratio: Calculate the cross ratio for the points 22, 1+i1 + i, 11, and 1+i1 + i using the formula (z1−z3)(z2−z4)/(z1−z4)(z2−z3)(z_1 - z_3)(z_2 - z_4) / (z_1 - z_4)(z_2 - z_3).\newlineCross ratio = (2−1)(1+i−1+i)/(2−1+i)(1+i−1)(2 - 1)(1 + i - 1 + i) / (2 - 1 + i)(1 + i - 1)
  2. Simplify Numerator and Denominator: Simplify the numerator and the denominator.\newlineCross ratio = (1)(2i)/(1+i)(i)(1)(2i) / (1 + i)(i)
  3. Multiply to Eliminate Complex Number: Multiply the numerator and denominator to get rid of the complex number in the denominator.\newlineCross ratio = (2i)/(i2+i)(2i) / (i^2 + i)
  4. Simplify i2i^2 and Continue: Simplify i2i^2 to −1-1 and continue simplifying.\newlineCross ratio = (2i)/(−1+i)(2i) / (-1 + i)
  5. Multiply by Conjugate: Multiply the numerator and denominator by the conjugate of the denominator to make the denominator real.\newlineCross ratio = (2i)(−1−i)/((−1+i)(−1−i))(2i)(-1 - i) / ((-1 + i)(-1 - i))
  6. Perform Multiplication: Perform the multiplication in the numerator and the denominator.\newlineCross ratio = (−2i−2i2)(1−i2)\frac{(-2i - 2i^2)}{(1 - i^2)}
  7. Combine Like Terms: Simplify i2i^2 to −1-1 in the numerator and denominator.\newlineCross ratio = (−2i+2)/(1+1)(-2i + 2) / (1 + 1)
  8. Divide by Denominator: Combine like terms and simplify the denominator.\newlineCross ratio = (2−2i)/2(2 - 2i) / 2
  9. Identify Conformal Points: Divide both terms in the numerator by the denominator.\newlineCross ratio = 1−i1 - i
  10. Set Derivative Equal to Zero: Now, identify the points where the mapping f(z)=12⋅(z+1z)f(z) = \frac{1}{2} \cdot (z + \frac{1}{z}) is conformal. Conformal mappings preserve angles, so we need to find where the derivative of f(z)f(z) does not equal zero or infinity. f′(z)=12⋅(1−1z2)f'(z) = \frac{1}{2} \cdot (1 - \frac{1}{z^2})
  11. Simplify by Multiplying: Set the derivative equal to zero to find where the mapping is not conformal.\newline0=12×(1−1z2)0 = \frac{1}{2} \times (1 - \frac{1}{z^2})
  12. Add 1z2\frac{1}{z^2}: Multiply both sides by 22 to simplify.\newline0=1−1z20 = 1 - \frac{1}{z^2}
  13. Take Square Root: Add 1/z21/z^2 to both sides.\newline1/z2=11/z^2 = 1
  14. Solve for z: Take the square root of both sides.\newline1z=±1\frac{1}{z} = \pm1
  15. Mapping Not Conformal: Take the reciprocal to solve for zz.z=±1z = \pm 1
  16. Mapping Not Conformal: Take the reciprocal to solve for zz. z=±1z = \pm 1 The mapping is not conformal at z=±1z = \pm 1.

More problems from Divide numbers written in scientific notation

Question
Factor 16a+72 16 a+72 to identify the equivalent expressions.\newlineChoose 22 answers:\newlineA 4(4a+18) 4(4 a+18) \newlineB 8(2a+9) 8(2 a+9) \newlineC 2(8+36a) 2(8+36 a) \newlineD 2(8a+72) 2(8 a+72)
Get tutor helpright-arrow

Posted 4 months ago

Question
Factor 30−6v−18w 30-6 v-18 w to identify the equivalent expressions.\newlineChoose 22 answers:\newlineA 2(15−12v) 2(15-12 v) \newlineB 3(10−3v+6w) 3(10-3 v+6 w) \newlinec 6(5−v−3w) 6(5-v-3 w) \newlineD 2(15−3v−9w) 2(15-3 v-9 w)
Get tutor helpright-arrow

Posted 4 months ago

Question
Factor 24m−12p+72 24 m-12 p+72 to identify the equivalent expressions.\newlineChoose 22 answers:\newlineA 6(4m+2p+12) 6(4 m+2 p+12) \newlineB 2(12m−6p+36) 2(12 m-6 p+36) \newlinec 12(2m−p+6) 12(2 m-p+6) \newlineD 24(m−12p+3) 24(m-12 p+3)
Get tutor helpright-arrow

Posted 4 months ago

Question
Factor 48−8x 48-8 x to identify the equivalent expressions.\newlineChoose 22 answers:\newlineA 3(16−8x) 3(16-8 x) \newlineB 2(24−4x) 2(24-4 x) \newlinec 8(6−x) 8(6-x) \newlineD 4(12−4x) 4(12-4 x)
Get tutor helpright-arrow

Posted 1 month ago

Question
Multiply and simplify the following complex numbers:\newline(−3−2i)⋅(−4+2i) (-3-2 i) \cdot(-4+2 i)
Get tutor helpright-arrow

Posted 3 months ago

Question
Multiply and simplify the following complex numbers:\newline(1−2i)⋅(4+i) (1-2 i) \cdot(4+i)
Get tutor helpright-arrow

Posted 3 months ago

Question
Multiply and simplify the following complex numbers:\newline(−1+4i)⋅(4−3i) (-1+4 i) \cdot(4-3 i)
Get tutor helpright-arrow

Posted 3 months ago

Question
Multiply and simplify the following complex numbers:\newline(−4−4i)⋅(−5−3i) (-4-4 i) \cdot(-5-3 i)
Get tutor helpright-arrow

Posted 3 months ago

Question
Multiply and simplify the following complex numbers:\newline(1+5i)⋅(−3−i) (1+5 i) \cdot(-3-i)
Get tutor helpright-arrow

Posted 3 months ago

Question
Multiply and simplify the following complex numbers:\newline(−2+2i)⋅(5+5i) (-2+2 i) \cdot(5+5 i)
Get tutor helpright-arrow

Posted 3 months ago

View all (41)