Where Does the Sign of n Matter in Palka's Examples?

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In summary, Peter is asking for help understanding two examples from Bruce Palka's book, An Introduction to Complex Function Theory. Peter is puzzled by the fact that each step of the calculation appears to work regardless of the value of n, and he needs help understanding where in the calculations n becomes relevant. Palka provides a separate proof for n negative, explaining that the power series converges to 0 as n becomes negative.
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MHB
I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ...

I am focused on Chapter III: Analytic Functions, Section 1.2 Differentiation Rules ...

I need help with some aspects of Examples 1.1 and 1.2, Section 1.2, Chapter III ...

Examples 1.1 and 1.2, Section 1.2, Chapter III read as follows:
View attachment 9334
My questions regarding the above two examples from Palka are as follows:Question 1

Can someone please explain where in the calculations of Example 1.1 does the assumption of n being positive becomes relevant ...

I am puzzled because it appears that each of the steps of the calculation are true whether n is positive or negative ...

Question 2

Can someone please explain where in the calculations of Example 1.2 does the assumption of n being negative becomes relevant ...

I am puzzled because it appears that each of the steps of the argument/calculation are true whether n is positive or negative ...Hope someone can help ...

Help will be much appreciated ...

Peter

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• Palka - Examples 1.1 and 1.2, Chapter III .png
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In $$z^n- z_0^n= (z- z_0)(z^{n-1}+ z_0z^{n-2}+ \cdot\cdot\cdot+ z_0^{n-2}z+ z_0^{n-1})$$, with n positive, powers of z are decreasing and eventually become 0 in the $$z_0^{n-1}$$ term. If n is negative powers of z decreasing become more negative so do not eventually become 0. We get a infinite series. It might well be true that you could prove the desired statement using infinite series but that Palka wants to avoid the additional complications of infinite series (which may not have been introduced at this point). And since the proof given is only for n positive, a separate proof has to be given for n negative.

HallsofIvy said:
In $$z^n- z_0^n= (z- z_0)(z^{n-1}+ z_0z^{n-2}+ \cdot\cdot\cdot+ z_0^{n-2}z+ z_0^{n-1})$$, with n positive, powers of z are decreasing and eventually become 0 in the $$z_0^{n-1}$$ term. If n is negative powers of z decreasing become more negative so do not eventually become 0. We get a infinite series. It might well be true that you could prove the desired statement using infinite series but that Palka wants to avoid the additional complications of infinite series (which may not have been introduced at this point). And since the proof given is only for n positive, a separate proof has to be given for n negative.

Thanks for the help, HallsofIvy ...

Peter

1. What are complex derivatives?

Complex derivatives are mathematical tools used in calculus to find the rate of change of a function with respect to a complex variable. They are similar to ordinary derivatives, but instead of using real numbers, they use complex numbers.

2. How are complex derivatives calculated?

Complex derivatives are calculated using the same rules as ordinary derivatives, but with complex numbers. This includes the power rule, product rule, quotient rule, and chain rule. However, the algebraic manipulations may be more complex due to the presence of imaginary numbers.

3. What are some common examples of complex derivatives?

Some common examples of complex derivatives include the derivative of a complex polynomial, the derivative of a complex exponential function, and the derivative of a complex trigonometric function. These derivatives are used in various fields of science and engineering, such as electromagnetism and quantum mechanics.

4. How are complex derivatives used in real-world applications?

Complex derivatives are used in many real-world applications, such as in electrical engineering to analyze alternating current circuits, in physics to study the behavior of quantum particles, and in economics to model financial markets. They are also used in image processing and signal analysis.

5. What is the significance of understanding complex derivatives?

Understanding complex derivatives is essential for advanced mathematical and scientific studies. They are used in many fields to solve complex problems and model real-world phenomena. Additionally, they provide a deeper understanding of the behavior of functions and their rates of change, which can lead to new discoveries and advancements in various fields of science and technology.

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