What does the notation [a,z] represent in the integral and supremum?

Conditions

• The points a and z are in the domain D.
• The straight line segment connecting a and z lies entirely within D.

Reasoning, step by step

1. Identify the points a and z in the complex plane.
2. Recognize that $[a,z]$ denotes the geometric line segment between these two points.
3. Understand that in the integral $\int_{[a,z]} f(w) dw$, $[a,z]$ specifies the path along which the function f is integrated.
4. Understand that in the expression $\sup_{w \in [a,z]} |f(w) - f(a)|$, $[a,z]$ specifies the set of points over which the maximum deviation of f from $f(a)$ is evaluated.

Example

The board writes $\int_{[a,z]} f(w)dw$ and $\sup_{w \in [a,z]} |f(w)-f(a)|$. The diagram shows a straight line segment connecting a and z inside the domain D. The lecturer refers to 'the line segment from a to z and its length'.

Common misconceptions

• Confusing the notation $[a,z]$ with an interval of real numbers.
• Assuming $[a,z]$ represents the distance between a and z, rather than the geometric path itself.

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Explore next

• Why can we use a straight line segment to calculate $F(z) - F(a)$ in the difference quotient?
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