Complex Contour Integral Calculator
Complex Contour Integral Calculator computes the contour integral \\( \oint_C f(z) \, dz \\) for a complex function \\( f(z) \\) over a parameterized contour, displaying steps with MathJax.
Complex Contour Integral Calculator
Complex Contour Integral Calculator computes the contour integral \\( \oint_C f(z) \, dz \\) for a complex function \\( f(z) \\) over a parameterized contour \\( z(t) \\), \\( t \in [a, b] \\). Input the function, contour parameterization, and parameter bounds to see the computed integral with MathJax-rendered steps. Results are copyable, with sharing and embedding options for complex analysis students.
Example 1: Contour Integral over Unit Circle
Function: \\( f(z) = \frac{1}{z} \\).
Contour: \\( z(t) = \cos(t) + i \sin(t) \\), \\( t \in [0, 2\pi] \\).
Step 1: Define the function and contour.
\\( f(z) = \frac{1}{z} \\), \\( z(t) = e^{it} \\).
Step 2: Compute the derivative.
\\( z'(t) = i e^{it} \\).
Step 3: Set up the integral.
\\( \oint_C f(z) \, dz = \int_0^{2\pi} f(z(t)) z'(t) \, dt = \int_0^{2\pi} \frac{1}{e^{it}} \cdot i e^{it} \, dt = \int_0^{2\pi} i \, dt \\).
Step 4: Evaluate the integral.
\\( \int_0^{2\pi} i \, dt = i [t]_0^{2\pi} = i (2\pi – 0) = 2\pi i \\).
Step 5: Conclusion.
The contour integral is \\( 2\pi i \\).
Example 2: Contour Integral over Line Segment
Function: \\( f(z) = z^2 \\).
Contour: \\( z(t) = t \\), \\( t \in [0, 1] \\).
Step 1: Define the function and contour.
\\( f(z) = z^2 \\), \\( z(t) = t \\).
Step 2: Compute the derivative.
\\( z'(t) = 1 \\).
Step 3: Set up the integral.
\\( \oint_C f(z) \, dz = \int_0^1 f(z(t)) z'(t) \, dt = \int_0^1 t^2 \cdot 1 \, dt \\).
Step 4: Evaluate the integral.
\\( \int_0^1 t^2 \, dt = \left[ \frac{t^3}{3} \right]_0^1 = \frac{1}{3} \\).
Step 5: Conclusion.
The contour integral is \\( \frac{1}{3} \\).