To evaluate the given complex integral, we use the fundamental theorem of calculus for complex functions. The integral of \(\frac{{{\rm{1}}}}{{\rm{z}}}\) with respect to \(\rm{z}\) is the complex logarithm, \(\rm{Ln(z)}\).

The definite integral can be calculated as follows:

\(\mathop \smallint \limits_5^{3{\rm{i}}} \frac{{{\rm{dz}}}}{{\rm{z}}} = \left[ {{\rm{Ln}}({\rm{z}})} \right]_5^{3{\rm{i}}} = {\rm{Ln}}(3{\rm{i}}) - {\rm{Ln}}(5)\)

We use the principal value of the complex logarithm, which is defined as:

\(\rm{Ln(z) = ln|z| + i Arg(z)}\)

where \(\rm{ln|z|}\) is the natural logarithm of the magnitude of \(\rm{z}\), and \(\rm{Arg(z)}\) is the principal argument of \(\rm{z}\), typically in the range \(\rm{(-\pi, \pi]}\).

Evaluating Ln(z) at Upper Limit (3i)

For the upper limit, \(\rm{z = 3i}\):

• Magnitude: \(\rm{|3i| = \sqrt{0^2 + 3^2} = 3}\)
• Principal Argument: Since \(\rm{3i}\) lies on the positive imaginary axis, its argument is \(\rm{Arg(3i) = \frac{\pi}{2}}\).

So, \(\rm{Ln(3i) = ln(3) + i\frac{\pi}{2}}\)

Evaluating Ln(z) at Lower Limit (5)

For the lower limit, \(\rm{z = 5}\):

• Magnitude: \(\rm{|5| = 5}\)
• Principal Argument: Since \(\rm{5}\) lies on the positive real axis, its argument is \(\rm{Arg(5) = 0}\).

So, \(\rm{Ln(5) = ln(5) + i(0) = ln(5)}\)

Calculating the Definite Complex Integral

Now, substitute these values into the integral formula:

\(\mathop \smallint \limits_5^{3{\rm{i}}} \frac{{{\rm{dz}}}}{{\rm{z}}} = {\rm{Ln}}(3{\rm{i}}) - {\rm{Ln}}(5) = \left( {{\rm{ln}}(3) + {\rm{i}}\frac{{\rm{\pi }}}{2}} \right) - {\rm{ln}}(5)\)

Separate the real and imaginary parts:

\(\mathop \smallint \limits_5^{3{\rm{i}}} \frac{{{\rm{dz}}}}{{\rm{z}}} = ({\rm{ln}}(3) - {\rm{ln}}(5)) + {\rm{i}}\frac{{\rm{\pi }}}{2}\)

Using the logarithm property \(\rm{ln(a) - ln(b) = ln(\frac{a}{b})}\):

\(\mathop \smallint \limits_5^{3{\rm{i}}} \frac{{{\rm{dz}}}}{{\rm{z}}} = {\rm{ln}}\left( {\frac{3}{5}} \right) + {\rm{i}}\frac{{\rm{\pi }}}{2}\)

Numerical Evaluation of the Integral

Now, we calculate the numerical values of the components:

• For the real part:

  The numerical values for the logarithms are:

  \(\rm{ln(3) \approx 1.0986}\)

  \(\rm{ln(5) \approx 1.6094}\)

  Using these values, the real part is calculated as:

  \(\rm{ln(5) - ln(3) \approx 1.6094 - 1.0986 \approx 0.5108}\)

  Rounding to three decimal places, the real part is approximately \(\rm{0.511}\). This corresponds to \(\rm{ln(\frac{5}{3})}\).

• For the imaginary part:

  The value of \(\rm{\pi}\) is approximately \(\rm{3.14159}\).

  \(\rm{\frac{\pi}{2} \approx \frac{3.14159}{2} \approx 1.57079}\)

  Rounding to two decimal places, the imaginary part is approximately \(\rm{1.57}\).

Combining the real and imaginary parts, the value of the integral is:

\(\mathop \smallint \limits_5^{3{\rm{i}}} \frac{{{\rm{dz}}}}{{\rm{z}}} \approx 0.511 + {\rm{i}}(1.57)\)

This result matches option 2 and 4.