Mixing
Analogous “Dark Sector” Trigonometric (and Hyperbolic) Functions
$begingroup$
Existing Definitions:
$$zeta(n)=sum_{k=1}^infty frac{ 1 }{k^n}$$
$$lambda(n)=sum_{k=1}^infty frac{ 1 }{(2k-1)^n}=frac{left(2^n-1right)}{2^n}zeta (n)$$
$$eta(n)=sum_{k=1}^infty frac{(-1)^{k-1}}{k^n}=left(1 -2^{1-n} right) zeta (n)$$
$$beta(n)=sum_{k=1}^infty frac{(-1)^{k-1}}{(2k-1)^n}$$
The existing well known trigonometric functions $csc(x)$, $sec(x)$, $tan(x)$ and $cot(x)$ in infinite series form are:
$$csc(x)=frac{1}{x}+2 sum _{k=1}^{infty } frac{eta(2 k) }{pi ^{2 k}};x^{2 k-1}$$
$$sec(x)=2sum _{k=1}^{infty } frac{ 2^{2 k-1} beta(2 k-1) }{pi ^{2 k-1}},x^{2 k-2}$$
$$tan(x)=2 sum _{k=1}^{infty } frac{2^{2 k} lambda(2 k) }{pi ^{2 k}},x^{2 k-1}$$
$$cot(x)=frac{1}{x}-2 sum _{k=1}^{infty } frac{ zeta (2 k)}{pi ^{2 k}},x^{2 k-1}$$
I am now going to define some analogous new definitions, using the odd Zeta constants and the even Beta constants, and postfix them with an "i":
$$text{csci}(x) =2 sum _{k=1}^{infty } frac{eta(2 k+1) }{pi ^{2 k+1}},x^{2 k}-frac{1}{x}+frac{2 log (2)}{pi }$$
$$text{seci}(x)=2sum _{k=1}^{infty } frac{ 2^{2 k}; beta(2 k) }{pi ^{2 k}},x^{2 k-1}$$
$$text{tani}(x)=2 sum _{k=1}^{infty } frac{2^{2 k+1} ;lambda(2 k+1) }{pi ^{2 k+1}},x^{2 k}+frac{2 log (2)}{pi }$$
$$text{coti}(x)=-2 sum _{k=1}^{infty } frac{ zeta (2 k+1)}{pi ^{2 k+1}},x^{2 k}-frac{1}{x}+frac{2 log (2)}{pi }$$
You could try the same analogy with "dark sector" hyperbolic functions. e.g.
$$text{sechi}(x)=2sum _{k=1}^{infty } frac{(-1)^{k-1} 2^{2 k}; beta(2 k) }{pi ^{2 k}},x^{2 k-1}$$
It is immediately apparent that there are certain similarities between normal trigonometry and "dark sector" trigonometry
For example $text{seci}(x)=text{csci}(x+frac{pi}{2})$, and
$text{csci}(x)=text{coti}(x/2)-text{coti}(x)$.
But also differences for example $text{tani}(x)$ is not equal to the inverse of $text{coti}(x)$. However both the inverses appear to lead to the same new function, that differs only by an inversion and a phase shift of $pi/2$.
Graph for $1/text{tani}(x)$
Graph for $1/text{coti}(x)$
Similar thing happens with the inverse of $1/text{csci}(x)$ and $1/text{seci}(x)$ to give $text{sini}(x)$ and $text{cosi}(x)$
Graph for $text{sini}(x)$
Graph for $text{cosi}(x)$
Examples of combining "dark sector" functions with normal trigonometric functions look quite interesting:
Graph for $text{cosi}(x)+cos(x)$
Graph for $text{cosi}(x)-cos(x)$
I've just sketched this structure using Mathematica before I waste too much time on it. There may be a better way of defining the four starting analogous functions: $text{csci}(x)$,$text{seci}(x)$,$text{tani}(x)$ and $text{coti}(x)$.
Does anyone know of attempts to develop what I call here "dark sector" trig or hyperbolic functions?
Does anyone recognise any of these functions and where they might have an application?
sequences-and-series trigonometry
asked Jan 27 at 14:31
James ArathoonJames Arathoon
1,618423
$endgroup$
add a comment |
$begingroup$
Existing Definitions:
$$zeta(n)=sum_{k=1}^infty frac{ 1 }{k^n}$$
$$lambda(n)=sum_{k=1}^infty frac{ 1 }{(2k-1)^n}=frac{left(2^n-1right)}{2^n}zeta (n)$$
$$eta(n)=sum_{k=1}^infty frac{(-1)^{k-1}}{k^n}=left(1 -2^{1-n} right) zeta (n)$$
$$beta(n)=sum_{k=1}^infty frac{(-1)^{k-1}}{(2k-1)^n}$$
The existing well known trigonometric functions $csc(x)$, $sec(x)$, $tan(x)$ and $cot(x)$ in infinite series form are:
$$csc(x)=frac{1}{x}+2 sum _{k=1}^{infty } frac{eta(2 k) }{pi ^{2 k}};x^{2 k-1}$$
$$sec(x)=2sum _{k=1}^{infty } frac{ 2^{2 k-1} beta(2 k-1) }{pi ^{2 k-1}},x^{2 k-2}$$
$$tan(x)=2 sum _{k=1}^{infty } frac{2^{2 k} lambda(2 k) }{pi ^{2 k}},x^{2 k-1}$$
$$cot(x)=frac{1}{x}-2 sum _{k=1}^{infty } frac{ zeta (2 k)}{pi ^{2 k}},x^{2 k-1}$$
I am now going to define some analogous new definitions, using the odd Zeta constants and the even Beta constants, and postfix them with an "i":
$$text{csci}(x) =2 sum _{k=1}^{infty } frac{eta(2 k+1) }{pi ^{2 k+1}},x^{2 k}-frac{1}{x}+frac{2 log (2)}{pi }$$
$$text{seci}(x)=2sum _{k=1}^{infty } frac{ 2^{2 k}; beta(2 k) }{pi ^{2 k}},x^{2 k-1}$$
$$text{tani}(x)=2 sum _{k=1}^{infty } frac{2^{2 k+1} ;lambda(2 k+1) }{pi ^{2 k+1}},x^{2 k}+frac{2 log (2)}{pi }$$
$$text{coti}(x)=-2 sum _{k=1}^{infty } frac{ zeta (2 k+1)}{pi ^{2 k+1}},x^{2 k}-frac{1}{x}+frac{2 log (2)}{pi }$$
You could try the same analogy with "dark sector" hyperbolic functions. e.g.
$$text{sechi}(x)=2sum _{k=1}^{infty } frac{(-1)^{k-1} 2^{2 k}; beta(2 k) }{pi ^{2 k}},x^{2 k-1}$$
It is immediately apparent that there are certain similarities between normal trigonometry and "dark sector" trigonometry
For example $text{seci}(x)=text{csci}(x+frac{pi}{2})$, and
$text{csci}(x)=text{coti}(x/2)-text{coti}(x)$.
But also differences for example $text{tani}(x)$ is not equal to the inverse of $text{coti}(x)$. However both the inverses appear to lead to the same new function, that differs only by an inversion and a phase shift of $pi/2$.
Graph for $1/text{tani}(x)$
Graph for $1/text{coti}(x)$
Similar thing happens with the inverse of $1/text{csci}(x)$ and $1/text{seci}(x)$ to give $text{sini}(x)$ and $text{cosi}(x)$
Graph for $text{sini}(x)$
Graph for $text{cosi}(x)$
Examples of combining "dark sector" functions with normal trigonometric functions look quite interesting:
Graph for $text{cosi}(x)+cos(x)$
Graph for $text{cosi}(x)-cos(x)$
I've just sketched this structure using Mathematica before I waste too much time on it. There may be a better way of defining the four starting analogous functions: $text{csci}(x)$,$text{seci}(x)$,$text{tani}(x)$ and $text{coti}(x)$.
Does anyone know of attempts to develop what I call here "dark sector" trig or hyperbolic functions?
Does anyone recognise any of these functions and where they might have an application?
sequences-and-series trigonometry
asked Jan 27 at 14:31
James ArathoonJames Arathoon
1,618423
$endgroup$
1
$begingroup$
This is really interesting (and beautiful) ! $+1$ fur sure and I shall spend time working this post. Thanks for posting such a nice work. Cheers.
$endgroup$
– Claude Leibovici
Jan 27 at 14:44
$begingroup$
@ClaudeLeibovici: Thanks. By the way the constant $frac{2 log 2}{pi}$ is actually $frac{2, eta(1)}{pi}$ which can't be calculated using the Mathematica friendly definition for $eta(n)$ in terms of the Zeta Function I used at the top of the page, where $n>1$.
$endgroup$
– James Arathoon
Jan 28 at 16:33
$begingroup$
Could you explain why you call these functions "dark sector" functions? Also: are you looking for closed forms? I'm not exactly sure what you're asking
$endgroup$
– clathratus
Jan 29 at 1:13
2
$begingroup$
@clathratus: I am interested in the overall structure, which you don't see the possibility of unless you approach infinite series using zeta and beta constants. I'm calling it the "dark sector" because I know some sort of limited analogical structure to trigonometry can be formulated, but I don't see more than a rough outline. As an example of my lack of understanding, differentiating tani[x] produces a function that is similar to but not the same as seci[x]^2 calculated in Mathematica directly. This may mean something or nothing I have no idea.
$endgroup$
– James Arathoon
Jan 29 at 1:49
add a comment |
$begingroup$
Existing Definitions:
$$zeta(n)=sum_{k=1}^infty frac{ 1 }{k^n}$$
$$lambda(n)=sum_{k=1}^infty frac{ 1 }{(2k-1)^n}=frac{left(2^n-1right)}{2^n}zeta (n)$$
$$eta(n)=sum_{k=1}^infty frac{(-1)^{k-1}}{k^n}=left(1 -2^{1-n} right) zeta (n)$$
$$beta(n)=sum_{k=1}^infty frac{(-1)^{k-1}}{(2k-1)^n}$$
The existing well known trigonometric functions $csc(x)$, $sec(x)$, $tan(x)$ and $cot(x)$ in infinite series form are:
$$csc(x)=frac{1}{x}+2 sum _{k=1}^{infty } frac{eta(2 k) }{pi ^{2 k}};x^{2 k-1}$$
$$sec(x)=2sum _{k=1}^{infty } frac{ 2^{2 k-1} beta(2 k-1) }{pi ^{2 k-1}},x^{2 k-2}$$
$$tan(x)=2 sum _{k=1}^{infty } frac{2^{2 k} lambda(2 k) }{pi ^{2 k}},x^{2 k-1}$$
$$cot(x)=frac{1}{x}-2 sum _{k=1}^{infty } frac{ zeta (2 k)}{pi ^{2 k}},x^{2 k-1}$$
I am now going to define some analogous new definitions, using the odd Zeta constants and the even Beta constants, and postfix them with an "i":
$$text{csci}(x) =2 sum _{k=1}^{infty } frac{eta(2 k+1) }{pi ^{2 k+1}},x^{2 k}-frac{1}{x}+frac{2 log (2)}{pi }$$
$$text{seci}(x)=2sum _{k=1}^{infty } frac{ 2^{2 k}; beta(2 k) }{pi ^{2 k}},x^{2 k-1}$$
$$text{tani}(x)=2 sum _{k=1}^{infty } frac{2^{2 k+1} ;lambda(2 k+1) }{pi ^{2 k+1}},x^{2 k}+frac{2 log (2)}{pi }$$
$$text{coti}(x)=-2 sum _{k=1}^{infty } frac{ zeta (2 k+1)}{pi ^{2 k+1}},x^{2 k}-frac{1}{x}+frac{2 log (2)}{pi }$$
You could try the same analogy with "dark sector" hyperbolic functions. e.g.
$$text{sechi}(x)=2sum _{k=1}^{infty } frac{(-1)^{k-1} 2^{2 k}; beta(2 k) }{pi ^{2 k}},x^{2 k-1}$$
It is immediately apparent that there are certain similarities between normal trigonometry and "dark sector" trigonometry
For example $text{seci}(x)=text{csci}(x+frac{pi}{2})$, and
$text{csci}(x)=text{coti}(x/2)-text{coti}(x)$.
But also differences for example $text{tani}(x)$ is not equal to the inverse of $text{coti}(x)$. However both the inverses appear to lead to the same new function, that differs only by an inversion and a phase shift of $pi/2$.
Graph for $1/text{tani}(x)$
Graph for $1/text{coti}(x)$
Similar thing happens with the inverse of $1/text{csci}(x)$ and $1/text{seci}(x)$ to give $text{sini}(x)$ and $text{cosi}(x)$
Graph for $text{sini}(x)$
Graph for $text{cosi}(x)$
Examples of combining "dark sector" functions with normal trigonometric functions look quite interesting:
Graph for $text{cosi}(x)+cos(x)$
Graph for $text{cosi}(x)-cos(x)$
I've just sketched this structure using Mathematica before I waste too much time on it. There may be a better way of defining the four starting analogous functions: $text{csci}(x)$,$text{seci}(x)$,$text{tani}(x)$ and $text{coti}(x)$.
Does anyone know of attempts to develop what I call here "dark sector" trig or hyperbolic functions?
Does anyone recognise any of these functions and where they might have an application?
sequences-and-series trigonometry
asked Jan 27 at 14:31
James ArathoonJames Arathoon
1,618423
$endgroup$
Existing Definitions:
$$zeta(n)=sum_{k=1}^infty frac{ 1 }{k^n}$$
$$lambda(n)=sum_{k=1}^infty frac{ 1 }{(2k-1)^n}=frac{left(2^n-1right)}{2^n}zeta (n)$$
$$eta(n)=sum_{k=1}^infty frac{(-1)^{k-1}}{k^n}=left(1 -2^{1-n} right) zeta (n)$$
$$beta(n)=sum_{k=1}^infty frac{(-1)^{k-1}}{(2k-1)^n}$$
The existing well known trigonometric functions $csc(x)$, $sec(x)$, $tan(x)$ and $cot(x)$ in infinite series form are:
$$csc(x)=frac{1}{x}+2 sum _{k=1}^{infty } frac{eta(2 k) }{pi ^{2 k}};x^{2 k-1}$$
$$sec(x)=2sum _{k=1}^{infty } frac{ 2^{2 k-1} beta(2 k-1) }{pi ^{2 k-1}},x^{2 k-2}$$
$$tan(x)=2 sum _{k=1}^{infty } frac{2^{2 k} lambda(2 k) }{pi ^{2 k}},x^{2 k-1}$$
$$cot(x)=frac{1}{x}-2 sum _{k=1}^{infty } frac{ zeta (2 k)}{pi ^{2 k}},x^{2 k-1}$$
I am now going to define some analogous new definitions, using the odd Zeta constants and the even Beta constants, and postfix them with an "i":
$$text{csci}(x) =2 sum _{k=1}^{infty } frac{eta(2 k+1) }{pi ^{2 k+1}},x^{2 k}-frac{1}{x}+frac{2 log (2)}{pi }$$
$$text{seci}(x)=2sum _{k=1}^{infty } frac{ 2^{2 k}; beta(2 k) }{pi ^{2 k}},x^{2 k-1}$$
$$text{tani}(x)=2 sum _{k=1}^{infty } frac{2^{2 k+1} ;lambda(2 k+1) }{pi ^{2 k+1}},x^{2 k}+frac{2 log (2)}{pi }$$
$$text{coti}(x)=-2 sum _{k=1}^{infty } frac{ zeta (2 k+1)}{pi ^{2 k+1}},x^{2 k}-frac{1}{x}+frac{2 log (2)}{pi }$$
You could try the same analogy with "dark sector" hyperbolic functions. e.g.
$$text{sechi}(x)=2sum _{k=1}^{infty } frac{(-1)^{k-1} 2^{2 k}; beta(2 k) }{pi ^{2 k}},x^{2 k-1}$$
It is immediately apparent that there are certain similarities between normal trigonometry and "dark sector" trigonometry
For example $text{seci}(x)=text{csci}(x+frac{pi}{2})$, and
$text{csci}(x)=text{coti}(x/2)-text{coti}(x)$.
But also differences for example $text{tani}(x)$ is not equal to the inverse of $text{coti}(x)$. However both the inverses appear to lead to the same new function, that differs only by an inversion and a phase shift of $pi/2$.
Graph for $1/text{tani}(x)$
Graph for $1/text{coti}(x)$
Similar thing happens with the inverse of $1/text{csci}(x)$ and $1/text{seci}(x)$ to give $text{sini}(x)$ and $text{cosi}(x)$
Graph for $text{sini}(x)$
Graph for $text{cosi}(x)$
Examples of combining "dark sector" functions with normal trigonometric functions look quite interesting:
Graph for $text{cosi}(x)+cos(x)$
Graph for $text{cosi}(x)-cos(x)$
I've just sketched this structure using Mathematica before I waste too much time on it. There may be a better way of defining the four starting analogous functions: $text{csci}(x)$,$text{seci}(x)$,$text{tani}(x)$ and $text{coti}(x)$.
Does anyone know of attempts to develop what I call here "dark sector" trig or hyperbolic functions?
Does anyone recognise any of these functions and where they might have an application?
sequences-and-series trigonometry
sequences-and-series trigonometry
asked Jan 27 at 14:31
James ArathoonJames Arathoon
1,618423
asked Jan 27 at 14:31
James ArathoonJames Arathoon
1,618423
edited Jan 27 at 16:35
James Arathoon
asked Jan 27 at 14:31
James ArathoonJames Arathoon
1,618423
asked Jan 27 at 14:31
James ArathoonJames Arathoon
1,618423
asked Jan 27 at 14:31
James ArathoonJames Arathoon
1,618423
1,618423
1
$begingroup$
This is really interesting (and beautiful) ! $+1$ fur sure and I shall spend time working this post. Thanks for posting such a nice work. Cheers.
$endgroup$
– Claude Leibovici
Jan 27 at 14:44
$begingroup$
@ClaudeLeibovici: Thanks. By the way the constant $frac{2 log 2}{pi}$ is actually $frac{2, eta(1)}{pi}$ which can't be calculated using the Mathematica friendly definition for $eta(n)$ in terms of the Zeta Function I used at the top of the page, where $n>1$.
$endgroup$
– James Arathoon
Jan 28 at 16:33
$begingroup$
Could you explain why you call these functions "dark sector" functions? Also: are you looking for closed forms? I'm not exactly sure what you're asking
$endgroup$
– clathratus
Jan 29 at 1:13
2
$begingroup$
@clathratus: I am interested in the overall structure, which you don't see the possibility of unless you approach infinite series using zeta and beta constants. I'm calling it the "dark sector" because I know some sort of limited analogical structure to trigonometry can be formulated, but I don't see more than a rough outline. As an example of my lack of understanding, differentiating tani[x] produces a function that is similar to but not the same as seci[x]^2 calculated in Mathematica directly. This may mean something or nothing I have no idea.
$endgroup$
– James Arathoon
Jan 29 at 1:49
add a comment |
1
$begingroup$
This is really interesting (and beautiful) ! $+1$ fur sure and I shall spend time working this post. Thanks for posting such a nice work. Cheers.
$endgroup$
– Claude Leibovici
Jan 27 at 14:44
$begingroup$
@ClaudeLeibovici: Thanks. By the way the constant $frac{2 log 2}{pi}$ is actually $frac{2, eta(1)}{pi}$ which can't be calculated using the Mathematica friendly definition for $eta(n)$ in terms of the Zeta Function I used at the top of the page, where $n>1$.
$endgroup$
– James Arathoon
Jan 28 at 16:33
$begingroup$
Could you explain why you call these functions "dark sector" functions? Also: are you looking for closed forms? I'm not exactly sure what you're asking
$endgroup$
– clathratus
Jan 29 at 1:13
2
$begingroup$
@clathratus: I am interested in the overall structure, which you don't see the possibility of unless you approach infinite series using zeta and beta constants. I'm calling it the "dark sector" because I know some sort of limited analogical structure to trigonometry can be formulated, but I don't see more than a rough outline. As an example of my lack of understanding, differentiating tani[x] produces a function that is similar to but not the same as seci[x]^2 calculated in Mathematica directly. This may mean something or nothing I have no idea.
$endgroup$
– James Arathoon
Jan 29 at 1:49
1
1
$begingroup$
This is really interesting (and beautiful) ! $+1$ fur sure and I shall spend time working this post. Thanks for posting such a nice work. Cheers.
$endgroup$
– Claude Leibovici
Jan 27 at 14:44
$begingroup$
This is really interesting (and beautiful) ! $+1$ fur sure and I shall spend time working this post. Thanks for posting such a nice work. Cheers.
$endgroup$
– Claude Leibovici
Jan 27 at 14:44
$begingroup$
@ClaudeLeibovici: Thanks. By the way the constant $frac{2 log 2}{pi}$ is actually $frac{2, eta(1)}{pi}$ which can't be calculated using the Mathematica friendly definition for $eta(n)$ in terms of the Zeta Function I used at the top of the page, where $n>1$.
$endgroup$
– James Arathoon
Jan 28 at 16:33
$begingroup$
@ClaudeLeibovici: Thanks. By the way the constant $frac{2 log 2}{pi}$ is actually $frac{2, eta(1)}{pi}$ which can't be calculated using the Mathematica friendly definition for $eta(n)$ in terms of the Zeta Function I used at the top of the page, where $n>1$.
$endgroup$
– James Arathoon
Jan 28 at 16:33
$begingroup$
Could you explain why you call these functions "dark sector" functions? Also: are you looking for closed forms? I'm not exactly sure what you're asking
$endgroup$
– clathratus
Jan 29 at 1:13
$begingroup$
Could you explain why you call these functions "dark sector" functions? Also: are you looking for closed forms? I'm not exactly sure what you're asking
$endgroup$
– clathratus
Jan 29 at 1:13
2
2
$begingroup$
@clathratus: I am interested in the overall structure, which you don't see the possibility of unless you approach infinite series using zeta and beta constants. I'm calling it the "dark sector" because I know some sort of limited analogical structure to trigonometry can be formulated, but I don't see more than a rough outline. As an example of my lack of understanding, differentiating tani[x] produces a function that is similar to but not the same as seci[x]^2 calculated in Mathematica directly. This may mean something or nothing I have no idea.
$endgroup$
– James Arathoon
Jan 29 at 1:49
$begingroup$
@clathratus: I am interested in the overall structure, which you don't see the possibility of unless you approach infinite series using zeta and beta constants. I'm calling it the "dark sector" because I know some sort of limited analogical structure to trigonometry can be formulated, but I don't see more than a rough outline. As an example of my lack of understanding, differentiating tani[x] produces a function that is similar to but not the same as seci[x]^2 calculated in Mathematica directly. This may mean something or nothing I have no idea.
$endgroup$
– James Arathoon
Jan 29 at 1:49
add a comment |
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$begingroup$
This is really interesting (and beautiful) ! $+1$ fur sure and I shall spend time working this post. Thanks for posting such a nice work. Cheers.
$endgroup$
– Claude Leibovici
Jan 27 at 14:44
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@ClaudeLeibovici: Thanks. By the way the constant $frac{2 log 2}{pi}$ is actually $frac{2, eta(1)}{pi}$ which can't be calculated using the Mathematica friendly definition for $eta(n)$ in terms of the Zeta Function I used at the top of the page, where $n>1$.
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– James Arathoon
Jan 28 at 16:33
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Could you explain why you call these functions "dark sector" functions? Also: are you looking for closed forms? I'm not exactly sure what you're asking
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– clathratus
Jan 29 at 1:13
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@clathratus: I am interested in the overall structure, which you don't see the possibility of unless you approach infinite series using zeta and beta constants. I'm calling it the "dark sector" because I know some sort of limited analogical structure to trigonometry can be formulated, but I don't see more than a rough outline. As an example of my lack of understanding, differentiating tani[x] produces a function that is similar to but not the same as seci[x]^2 calculated in Mathematica directly. This may mean something or nothing I have no idea.
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– James Arathoon
Jan 29 at 1:49