Derivatives on HP 42S
08-26-2018, 04:54 AM
Post: #19
 Thomas Klemm Senior Member Posts: 1,534 Joined: Dec 2013
RE: Derivatives on HP 42S
(08-25-2018 09:20 PM)Albert Chan Wrote:  Can you explain the word analytic ?

Consider the complex valued function $$w=z^2$$.
You can calculate both the real and imaginary part of $$w=u+iv$$:

\begin{aligned} u &= x^2-y^2 \\ v &= 2xy \end{aligned}

But these functions $$u(x, y)$$ and $$v(x, y)$$ are not independent.
Instead the Cauchyâ€“Riemann equations hold true:

\begin{aligned} \frac {\partial u}{\partial x}&=\frac{\partial v}{\partial y} \\ \frac {\partial u}{\partial y}&=-\frac{\partial v}{\partial x} \end{aligned}

And indeed:

\begin{aligned} u_x &=2x=v_y \\ u_y &=-2y=-v_x \end{aligned}

Thus this function is analytic.

However this function isn't:

\begin{aligned} u &= x^2+x-y^2 \\ v &= 2xy \end{aligned}

Because $$u_x=2x+1$$ but still $$v_y=2x$$.

For the other function that I mentioned, e.g. $$\Re[z]$$ we have:

\begin{aligned} u &= x \\ v &= 0 \end{aligned}

This isn't analytic since $$u_x=1$$ but $$v_y=0$$.

In short: If you define the function in terms of $$z$$ the function is most probably analytic. However if you try to stitch together a complex function based on $$x$$ and $$y$$ chances are high that it's not analytic.

Quote:Is f(x) = x^(1/3) an analytic function ?

Yes. Its derivative is:

$$\frac{d}{dz}\left(\sqrt[3]{z}\right)=\frac{1}{3z^{\frac{2}{3}}}$$

Quote:If x is complex, is it true that f(x) same as -f(-x) ?

No. Consider $$f(z)=z^2$$. Here we have $$f(-z)=f(z)$$.

Cheers
Thomas
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 Messages In This Thread Derivatives on HP 42S - lrdheat - 08-20-2018, 03:03 AM RE: Derivatives on HP 42S - Thomas Klemm - 08-20-2018, 04:38 AM RE: Derivatives on HP 42S - Thomas Klemm - 08-20-2018, 07:43 AM RE: Derivatives on HP 42S - Albert Chan - 08-20-2018, 11:54 PM RE: Derivatives on HP 42S - lrdheat - 08-20-2018, 10:57 PM RE: Derivatives on HP 42S - Thomas Klemm - 08-20-2018, 11:43 PM RE: Derivatives on HP 42S - Thomas Klemm - 08-21-2018, 12:34 AM RE: Derivatives on HP 42S - Thomas Klemm - 08-21-2018, 01:35 AM RE: Derivatives on HP 42S - lrdheat - 08-21-2018, 02:24 AM RE: Derivatives on HP 42S - Thomas Klemm - 08-21-2018, 06:14 AM RE: Derivatives on HP 42S - RMollov - 08-23-2018, 12:58 PM RE: Derivatives on HP 42S - lrdheat - 08-24-2018, 02:51 AM RE: Derivatives on HP 42S - Thomas Klemm - 08-24-2018, 05:52 AM RE: Derivatives on HP 42S - lrdheat - 08-25-2018, 05:19 PM RE: Derivatives on HP 42S - Albert Chan - 08-25-2018, 07:03 PM RE: Derivatives on HP 42S - Thomas Klemm - 08-25-2018, 06:05 PM RE: Derivatives on HP 42S - Thomas Klemm - 08-25-2018, 08:00 PM RE: Derivatives on HP 42S - Albert Chan - 08-25-2018, 09:20 PM RE: Derivatives on HP 42S - Thomas Klemm - 08-26-2018 04:54 AM RE: Derivatives on HP 42S - Thomas Okken - 08-26-2018, 01:54 PM RE: Derivatives on HP 42S - lrdheat - 08-26-2018, 04:47 PM RE: Derivatives on HP 42S - Albert Chan - 08-26-2018, 08:39 PM RE: Derivatives on HP 42S - Thomas Klemm - 08-26-2018, 08:00 PM RE: Derivatives on HP 42S - Albert Chan - 08-29-2018, 01:52 PM

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