Contour Integral or Complex Path Integral

Let's find contour integral of 1/Z where path of Z is a circle on complex plane. First, we need to define expression for the circle using a separate parameter variable T and add it to our CAL list. The expression should define both real and imaginary parts of the complex path.

So, we input the complex expression cos(T) + i * sin(T) and add it to the CAL list using the AddToList option. (Use the F1 option to add cos and sin functions and VAR option to add parameter var T)

After saving the expression for the path to CAL list, we clear the current expression box and enter the expression 1/Z and tap ≈∫ to integrate it with respect to var Z.

Select var Z as our integral var. By default, the app assumes the path of var to be real. Since we know that Z is a complex path, we select the option "Is var below complex?" and tap on "Complex Path..." to define the path of Z on complex plane.

We use the saved expression for complex path from the CAL list and specify parameter var as T that was used in the expression for path. 

We set the start of param var T to 0 and end to 6.283185307179 (2pi)

Proceed with integration and we get approximate result 2 π  i  as shown in the snapshot below.

Contour integral using spreadsheet data.

The app also provides the option to perform contour integral using a spreadsheet file data that defines both the values for the integrand and the complex path.

The column data in spreadsheet can have complex values in the form a+bi or a-bi

A sample data read by app from spreadsheet is shown below. The first column has the values for path and the second column has values for the integrand, both in complex form. 

After reading this spreadsheet in the app we simply enter the expression Y (By default mapped to column 2) and integrate. We select the integral var as X (By default mapped to column 1) and also select the option "Is var below complex?"

This will integrate all data in file for column 2 (var Y) with respect to path data in column 1 (var X).

(Please refer to help "Plot or integrate using file spreadsheet data" for details on how to integrate on file data using expressions)