Complex Taylor series

[DicoMico]

TASK:
Assuming a complex function f(z) can be expanded as a Taylor series around z=0, i.e.:

$$f(z)=\sum_{n=0}^{\infty}a_{n}z^n$$

Setting z=r*exp(i*theta), assuming a_n is real, find real part u(r, theta), imaginary part v(r,theta).

Comment the result, especially for r=1.


MY SOLUTION:
$$f(z)=\sum_{n=0}^{\infty}a_{n}r^n(e^{i\theta})^n = \sum_{n=0}^{\infty}a_{n}r^n(cosn\theta+isinn\theta)$$

$$u(r,\theta)=\sum_{n=0}^{\infty}a_{n}r^n(cosn\theta)$$
$$v(r,\theta)=\sum_{n=0}^{\infty}a_{n}r^n(sinn\theta)$$


So I am to comment on the result.. What is there to say? I have "decomposed" the complex series into two real series, but this is kindo given; I'm sure they're asking for another comment.. Ideas?

Is this an "optimal" solution BTW, or are there any other possibilities? Later on I'm asked to find real part of a expansion of ln.

 

[lippyka]

ANSWER:The result of decomposing a complex function into its real and imaginary parts is that it allows us to analyze the behavior of the function in terms of its real and imaginary components. When r=1, it means the function is evaluated at a point on the unit circle and therefore, the real part u(r,theta) gives us the x-coordinate of this point and the imaginary part v(r,theta) gives us the y-coordinate of this point. This can be used to graph the complex function's behavior and visualize how it changes with varying values of theta.