integrate:dz/(z^2+1)
where z is a complex number.....plz working
this is a standard form.
Let
z = tanθ
z^2+1 = sec^2θ
dz = sec^2θ dθ
Plug that in and see how simple it is.
You can also use
z = sinhθ
but that's less common.
what the meaning of that box
Hmmm. font problem, I guess. It's theta.
Let's use t
z = tan(t)
z^2+1 = sec^2(t)
dz = sec^2(t) dt
To integrate dz/(z^2+1), you can use the method of partial fractions. Here's how you can proceed:
Step 1: Factorize the denominator.
The denominator z^2+1 cannot be factored further since it is the sum of two squares and does not have any real roots.
Step 2: Express the integrand as a sum of partial fractions.
Since the denominator cannot be factored, we have an irreducible quadratic factor, which means the partial fraction decomposition will involve complex numbers. Let's express the integrand as the sum of two partial fractions as follows:
dz/(z^2+1) = A/(z-i) + B/(z+i)
Step 3: Find the values of A and B.
To find the values of A and B, we need to determine the numerators for each fraction. To do this, we can multiply through by the denominator and equate the coefficients of the corresponding powers of z:
dz = A(z+i) + B(z-i)
The equation holds for all values of z, so we can choose the values that will help us solve for A and B. Let's substitute z = -i and z = i:
At z = -i:
d(-i) = A(-i+i) + B(-i-i)
= 0 - 2Bi
At z = i:
d(i) = A(i+i) + B(i-i)
= 2Ai + 0
Comparing coefficients, we have:
0 = -2Bi => B = 0
2 = 2Ai => A = 1
So, the partial fraction decomposition is:
dz/(z^2+1) = 1/(z-i)
Step 4: Integrate the partial fractions.
Now we can integrate each partial fraction term separately:
∫ dz/(z^2+1) = ∫ 1/(z-i) dz
To integrate this, we can use the substitution method. Let u = z-i, then du = dz.
∫ 1/u du
= ln|u| + C
= ln|z-i| + C
Therefore, the final result of the integration is:
∫ dz/(z^2+1) = ln|z-i| + C
Note that C represents the constant of integration.