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Complex Analysis – The equation seems to violate the transitive properties

I have a function $ f (x) = sin (mx) cos (nx) $, I tried to rewrite this with Euler's formula.

$$
begin {align}
f (x) & = sin mx cos nx \
& = frac {e ^ {imx} – e ^ {- imx}} {2i} times frac {e ^ {inx} + e ^ {- inx}} {2} \
& = frac {1} {4i} left ( left (e ^ {imx} – e ^ {- imx} right)
times left (e ^ {inx} + e ^ {- inx} right) right) \
& = frac {1} {4 i} left (e ^ {ix (m + n)}
+ e ^ {ix (m-n)} – e ^ {ix (n-m)}
– e ^ {- ix (m + n)} right)
end
$$

However, something seems to be wrong in this last step. If I check the equality with sagemath:

x=var('x')
m=var('m')
n=var('n')
f(x)=sin(m*x)*cos(n*x)
g(x)=((exp(I*m*x)-exp(-I*m*x))*(exp(I*n*x)+exp(-I*n*x)))/(4*I)
h(x)=(exp(I*x*(m+n))+exp(I*x*(m-n))-exp(I*x*(n-m))-exp(-I*x*(m+n)))/(4*I)
bool(f(x)==g(x))
bool(g(x)==h(x))
bool(f(x)==h(x))

I get the result:

True
True
False

In other words, it tells me that
$$
sin mx cos nx = frac {1} {4i} left ( left (e ^ {imx} – e ^ {- imx} right)
times left (e ^ {inx} + e ^ {- inx} right) right)
$$
and
$$
frac {1} {4i} left ( left (e ^ {imx} – e ^ {- imx} right)
times left (e ^ {inx} + e ^ {- inx} right) = frac {1} {4 i} left (e ^ {ix (m + n)}
+ e ^ {ix (m-n)} – e ^ {ix (n-m)}
– e ^ {- ix (m + n)} right)
$$
but
$$
sin mx cos nx neq frac {1} {4 i} left (e ^ {ix (m + n)}
+ e ^ {ix (m-n)} – e ^ {ix (n-m)}
– e ^ {- ix (m + n)} right)
$$
How is that possible? Did I make a mistake in the last step?

gnome – Ubuntu 18.04.3 displays a broken complex problem

I'm new to this forum and I really need help repairing my Ubuntu desktop display. I know, this question has been asked so many times and I have already tried it, but so far no success.

I use Ubuntu LMS and shut down my computer normally, but Display Manager does not work anymore when I turned my computer on in the morning. First it was in the terminal Display Manager not working then I tried to fix and reinstall lightdm, gnome shell and gdm3, but nothing worked.

Now the system says Overlays : missing lower dir and it's about tty1, I can not install anything on my system and get a message

"You don't have enough space in /var/cache/apt/archives/.

I have more than 100 GB of free space and I also used sudo apt clean and sudo apt autoremove, but nothing works for me.

Please help me overcome this space problem and restore my desktop display.

How can you shade the region in the complex layer?

I'm having trouble understanding the following sentence:

{𝑧∈ℂ: | 𝑧-1 | = | 𝑧-𝑖 |}

How would I shadow the region?

How complex is it to find e ^ (A) for a Hermitian matrix A?

If A is a hermitic matrix of size NxN. of steps required to calculate e ^ (A).

Table columns with complex structures (fields underneath that may have fields underneath that look like nested ones) in some columns

This is for the web application. I'm looking for tabular UIs that have no nesting but can display a child row below a parent row. These child UIs can have more children, and so on. My limitation is not to use nesting.
Any hints or ideas about this? Enter image description here

Microsoft Excel – Complex conditional formatting

I am a CPA with a lot of Excel experience but can not seem to find an answer to this (my) question. Is there a way to apply conditional formatting to the following example:

EXAMPLE: I have the formula = sum (B4, C28, A32, B40) —> Is there a way to apply a conditional formatting rule to cells B4, C28, A32, and B40, which automatically highlights the cells in which they are are the formula used?

^ This would help enormously with a bank balance that I carry out. Instead of highlighting manually. There are many cells for which I would have to do this manually. I firmly believe that polling is as automated as possible (in fact, most of my polls are simply a drag-and-drop template that automatically identifies most polls).

Complex Geometry – Shows that a specific layer is set [of a continuous family of holomorphic maps] is locally path-connected

I work with a continuous function $ P: (0,1) times W to mathbb {C} ^ n $, Where $ W subset mathbb {C} ^ n $ is an open, relatively compact sphere centered at the origin. The map $ P $ meets the following conditions:

$ P (t, cdot) $ is holomorphic for everyone $ t in (0,1) $.
The sentence $ ( {t } times W) cap P ^ {- 1} (0) $ is finite and not empty for everyone $ t in (0,1) $, and
To let $ epsilon> 0 $, If $ (s, zeta_s) in P ^ {- 1} (0) $, then there is something $ Delta> 0 $ so for all $ t in (s- delta, s + delta) cap (0,1) $is there $ (t, zeta_t) in P ^ {- 1} (0) $ With $ | zeta_s – zeta_t | < epsilon $,

I would like to know if the above conditions give the set level $ P ^ {- 1} (0) $ Connected to a local path, and if not, what other assumptions might be needed to ensure this?

(I will give this problem a bit more context, though this may not be necessary: $ P $ is a & # 39; periodic chart & # 39; a certain continuous function $ Psi: (0,1) times W times D to mathbb {C} setminus {0 } $ Where $ D $ is a compact, continuous Riemann surface with boundary. That is, a non-vanishing holomorphic 1-form $ theta $ is fixed on $ D $and every component of $ P (t, zeta) $ is a period of the holomorphic 1-form $ Psi (t, zeta, cdot) theta $, Essentially, I am looking for a homotopy of holomorphic functions $ D $with disappearing periods).

I do not know if there is a general proposition that will allow us to draw this conclusion, or whether other conditions may have to be imposed.

cv.complex variables – The term holomorphic map in a complex space category

In the category of complex analytic space we sometimes also call the holomorphic map of morphism (eg Demailly book, etc.).
The name "holomorphic map" is somewhat confusing as you know that the holomorphic map is smooth in a complex diverse category. In the category of complex analytic space, however, the holomorphic map is, in my opinion, continuous only between its underlying topological space.
So, I want to know the following questions:

1, If a complex space is a complex manifold, does the notion of the holomorphic map agree with the holomorphic map in the complex manifold category?
2, Is the holomorphic map a smooth morphism (in schema theory)? (In the complex varied category, the holomorphic map is naturally smooth.)

PHP – HTML Complex Table

I need to create two pivot tables of these formats:

Fixed headings

Variable headings

In PHP / HTML related to a database in SQL.
In the first you have to allow modify the notes, that is, the the same box how do you know demonstrate the note must be variable, There are no significant problems with the first one (although I accept co-operation or suggestions), because all courses have the same number of headings, but the second has the same number The number of subjects varies by courseTherefore, a single code should be able to handle all topics. I do not know how to do that Interrogate of notes if I can not create a variable for every possible subject.
Where the year, division and orientation or shift are, a option as filter of the courses and being able to print different tables simply by changing the option.

I have very basic knowledge of PHP, SQL and HTML, but I managed to create the first table. I miss the options as a filter and check the queries well.
The second is the one that gets complicated.

Thank you very much.