plz i need full solution on this i haven't been taught b4...i was give this to solve plz plz help me
integral(dz/(|z|)²)
| z |² = z²
∫ dz / | z |² = ∫ dz / z² = ∫ z ⁻² dz
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Apply the power rule:
∫ xⁿ dx = xⁿ⁺¹ / ( n + 1 ) + C
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∫ z⁻² dz = z ⁻²⁺¹ / ( - 2 + 1 ) + C = z ⁻¹ / ( - 1 ) + C = - z ⁻¹ + C = - 1 / z + C
∫ dz / | z |² = - 1 / z + C
the question say i should use green's theorem
Are we dealing with a complex variable z, so that Green's Theorem states that
∫B(z,z*) dz = 2i ∫∫∂B/∂z* dx dy
So, we have B(z,z*) = zz*
and |z|^2 = x^2+y^2
?? Or are we just dealing with a vector function z = xi+yj+zk ??
yes complex variable z..pls ,show me how pls pls
yes yes
Sure, I can help you solve the integral ∫(dz/|z|²). Let's start with some explanations before providing the solution.
To solve this integral, we need to perform a change of variables. Let's consider the complex variable z, which can be written as z = x + iy, where x and y represent the real and imaginary parts of z, respectively.
In terms of x and y, the absolute value of z, denoted as |z|, can be expressed as:
|z| = √(x² + y²).
We can rewrite the integral in terms of x and y using the formula:
dz = dx + i dy.
Substituting these expressions into the integral, we have:
∫(dz/|z|²) = ∫(dx + i dy)/(√(x² + y²))².
Simplifying the denominator, we obtain:
∫(dx + i dy)/(x² + y²).
Now, we separate the integral into real and imaginary parts:
∫(dx + i dy)/(x² + y²) = ∫(dx)/(x² + y²) + i ∫(dy)/(x² + y²).
Let's solve each part separately:
1. Real Part:
∫(dx)/(x² + y²)
This integral can be evaluated using trigonometric substitution. The most common substitution is to let x = r cosθ and y = r sinθ, where r represents the distance from the origin (r = √(x² + y²)) and θ is the angle.
∫(dx)/(x² + y²) = ∫(dx)/(r²)
Using x = r cosθ, we can write dx = cosθ dr - r sinθ dθ. Substituting these into the integral, we have:
∫(dx)/(x² + y²) = ∫(cosθ dr - r sinθ dθ)/(r²)
= ∫(cosθ dr)/(r²) - ∫(sinθ/r) dθ.
The first integral can be evaluated as:
∫(cosθ dr)/(r²) = ∫(cosθ)(d(1/r))
= ∫(d(1/r)) cosθ
= (1/r) cosθ + C₁,
where C₁ is the constant of integration.
The second integral can be simplified using a trigonometric identity. Since sinθ/r = y/r² = y/(x² + y²), we have:
∫(sinθ/r) dθ = ∫(y/(x² + y²)) dθ
= ½ ln|x² + y²| + C₂,
where C₂ is the constant of integration.
Therefore, the real part of the integral is:
∫(dx)/(x² + y²) = (1/r) cosθ + C₁ - ½ ln|x² + y²| + C₂.
2. Imaginary Part:
∫(dy)/(x² + y²)
This integral can be evaluated similarly to the real part. Starting with y = r sinθ and x = r cosθ, we obtain:
∫(dy)/(x² + y²) = ∫(r sinθ)/(r²)
= ∫(sinθ)(d(1/r))
= ∫(d(1/r)) sinθ
= (1/r) sinθ + C₃,
where C₃ is the constant of integration.
Finally, the complete solution to the original integral is obtained by combining the real and imaginary parts:
∫(dz/|z|²) = ∫(dx + i dy)/(x² + y²)
= (1/r) cosθ + C₁ - ½ ln|x² + y²| + C₂ + i((1/r) sinθ + C₃).
Please note that the constants of integration (C₁, C₂, C₃) can vary depending on the specific initial conditions or constraints of the problem you are solving.