special functions defined as a solution to a trig equation with a variable parameter
Warning: This is kind of an open-ended question.
Often I have seen people interested in solving equations like $f(t)=g(t)$, where $f(t)$ is made up of trig functions and $g(t)$ is not. I was wondering if there have ever been studied special functions that map $g(t)$ (or its parameters) to solutions of $f(t)=g(t)$.
Here is a concrete example: For each $ain[1,infty)$ and each $xinmathbb{R}$ there is a unique solution (in $t$) to the equation
$$sin(t)=at-x.$$
We could therefore define a special function $text{ss}_a:mathbb{R}tomathbb{R}$ by letting each $text{ss}_a(x)$ denote the unique solution to the above.
Question 1. Have any such special functions been studied before? I don't mean that it has to be exactly of the above form, but just something like it.
EDIT:
In the comments section there was some question as to $text{ss}_a$ is well-defined, so here is a short proof. Write $sin(t)=at-x$ as $x=at-sin(t)$ Then $frac{dx}{dt}=a-cos(t)$ so that $x$ is bijective increasing as a function of $t$. Thus it has an inverse which is bijective increasing, and this inverse coincides with $text{ss}_a$.
In fact, by the Inverse Function Theorem we have that $text{ss}_a$ is continuously differentiable everywhere for $a>1$ with $text{ss}'_a=1/(a-cos(text{ss}_a(x))$. In case $a=1$ it is continuously differentiable except at $x=2pi n$, $ninmathbb{Z}$.
trigonometry special-functions
asked Jan 3 at 22:28
Ben W
1,995615
add a comment |
Warning: This is kind of an open-ended question.
Often I have seen people interested in solving equations like $f(t)=g(t)$, where $f(t)$ is made up of trig functions and $g(t)$ is not. I was wondering if there have ever been studied special functions that map $g(t)$ (or its parameters) to solutions of $f(t)=g(t)$.
Here is a concrete example: For each $ain[1,infty)$ and each $xinmathbb{R}$ there is a unique solution (in $t$) to the equation
$$sin(t)=at-x.$$
We could therefore define a special function $text{ss}_a:mathbb{R}tomathbb{R}$ by letting each $text{ss}_a(x)$ denote the unique solution to the above.
Question 1. Have any such special functions been studied before? I don't mean that it has to be exactly of the above form, but just something like it.
EDIT:
In the comments section there was some question as to $text{ss}_a$ is well-defined, so here is a short proof. Write $sin(t)=at-x$ as $x=at-sin(t)$ Then $frac{dx}{dt}=a-cos(t)$ so that $x$ is bijective increasing as a function of $t$. Thus it has an inverse which is bijective increasing, and this inverse coincides with $text{ss}_a$.
In fact, by the Inverse Function Theorem we have that $text{ss}_a$ is continuously differentiable everywhere for $a>1$ with $text{ss}'_a=1/(a-cos(text{ss}_a(x))$. In case $a=1$ it is continuously differentiable except at $x=2pi n$, $ninmathbb{Z}$.
trigonometry special-functions
asked Jan 3 at 22:28
Ben W
1,995615
Draw x as function of t and you will find that there are numerous (x,t) pairs that meet the equation. What exactly is there to study?
– Moti
Jan 4 at 1:44
add a comment |
Warning: This is kind of an open-ended question.
Often I have seen people interested in solving equations like $f(t)=g(t)$, where $f(t)$ is made up of trig functions and $g(t)$ is not. I was wondering if there have ever been studied special functions that map $g(t)$ (or its parameters) to solutions of $f(t)=g(t)$.
Here is a concrete example: For each $ain[1,infty)$ and each $xinmathbb{R}$ there is a unique solution (in $t$) to the equation
$$sin(t)=at-x.$$
We could therefore define a special function $text{ss}_a:mathbb{R}tomathbb{R}$ by letting each $text{ss}_a(x)$ denote the unique solution to the above.
Question 1. Have any such special functions been studied before? I don't mean that it has to be exactly of the above form, but just something like it.
EDIT:
In the comments section there was some question as to $text{ss}_a$ is well-defined, so here is a short proof. Write $sin(t)=at-x$ as $x=at-sin(t)$ Then $frac{dx}{dt}=a-cos(t)$ so that $x$ is bijective increasing as a function of $t$. Thus it has an inverse which is bijective increasing, and this inverse coincides with $text{ss}_a$.
In fact, by the Inverse Function Theorem we have that $text{ss}_a$ is continuously differentiable everywhere for $a>1$ with $text{ss}'_a=1/(a-cos(text{ss}_a(x))$. In case $a=1$ it is continuously differentiable except at $x=2pi n$, $ninmathbb{Z}$.
trigonometry special-functions
asked Jan 3 at 22:28
Ben W
1,995615
Warning: This is kind of an open-ended question.
Often I have seen people interested in solving equations like $f(t)=g(t)$, where $f(t)$ is made up of trig functions and $g(t)$ is not. I was wondering if there have ever been studied special functions that map $g(t)$ (or its parameters) to solutions of $f(t)=g(t)$.
Here is a concrete example: For each $ain[1,infty)$ and each $xinmathbb{R}$ there is a unique solution (in $t$) to the equation
$$sin(t)=at-x.$$
We could therefore define a special function $text{ss}_a:mathbb{R}tomathbb{R}$ by letting each $text{ss}_a(x)$ denote the unique solution to the above.
Question 1. Have any such special functions been studied before? I don't mean that it has to be exactly of the above form, but just something like it.
EDIT:
In the comments section there was some question as to $text{ss}_a$ is well-defined, so here is a short proof. Write $sin(t)=at-x$ as $x=at-sin(t)$ Then $frac{dx}{dt}=a-cos(t)$ so that $x$ is bijective increasing as a function of $t$. Thus it has an inverse which is bijective increasing, and this inverse coincides with $text{ss}_a$.
In fact, by the Inverse Function Theorem we have that $text{ss}_a$ is continuously differentiable everywhere for $a>1$ with $text{ss}'_a=1/(a-cos(text{ss}_a(x))$. In case $a=1$ it is continuously differentiable except at $x=2pi n$, $ninmathbb{Z}$.
trigonometry special-functions
trigonometry special-functions
asked Jan 3 at 22:28
Ben W
1,995615
asked Jan 3 at 22:28
Ben W
1,995615
edited Jan 4 at 3:09
asked Jan 3 at 22:28
Ben W
1,995615
asked Jan 3 at 22:28
Ben W
1,995615
asked Jan 3 at 22:28
Ben W
1,995615
1,995615
Draw x as function of t and you will find that there are numerous (x,t) pairs that meet the equation. What exactly is there to study?
– Moti
Jan 4 at 1:44
add a comment |
Draw x as function of t and you will find that there are numerous (x,t) pairs that meet the equation. What exactly is there to study?
– Moti
Jan 4 at 1:44
Draw x as function of t and you will find that there are numerous (x,t) pairs that meet the equation. What exactly is there to study?
– Moti
Jan 4 at 1:44
Draw x as function of t and you will find that there are numerous (x,t) pairs that meet the equation. What exactly is there to study?
– Moti
Jan 4 at 1:44
add a comment |
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Draw x as function of t and you will find that there are numerous (x,t) pairs that meet the equation. What exactly is there to study?
– Moti
Jan 4 at 1:44