The argument principle in complex analysis states that for a meromorphic nonconstant function ... on an open subset ... and a closed curve ... bounding a compact subset inside ... , we have ... , where ... and ... are the zeros and poles of ... inside ... , ... is the winding number of ... around ... , and ... is the multiplicity of ... at ... . In this Demonstration the zeros and poles of a chosen function lying in the disk around the origin of radius 2 are shown as red and blue points (multiple zeros are shown as single points). The path of integration initially consists of the red square; you can change its position and shape by dragging, adding, or subtracting the locator points. The value of the integral along the path (in the clockwise direction) is shown below. By unchecking the "closed curve" checkbox you can construct nonclosed curves by separating the two locators in the bottom-left corner.


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