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Derivatives on HP 42S
08-26-2018, 04:54 AM
Post: #19
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 Cauchy–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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