text: # Winding number of a radius dependent function?
  
  The question: Let $\gamma:[0,2\pi] \rightarrow \mathbb{C}, t \mapsto e^{int} + re^{imt}$, with $n,m \in \mathbb{Z}$ and $0<r \neq 1$. Calculate the winding number.
  
  My attempt: My first try was to calculate the integral $\operatorname{Ind}(\gamma,z) = \int_0^{2\pi} \frac{\gamma'(t)}{\gamma(t) - z}\,\mathrm dz$, but when I solved this integral by subsituting the denominator after some calculation I end up with a logarithm dependent on $r$ and $z$ divided by $2\pi i$ which is obviously wrong since the winding number should be an integer. I suppose there should be a solution by making a homotopy kind of argument without having to calculate the integral but I'm kinda stuck in here.
  
  Greetings a Peaceful Slosh
  
  Suppose that $0<r<1$. Then the loop $\gamma$ is homotopic to the loop $\gamma^\star\colon[0,2\pi]\longrightarrow\mathbb C$ defined by $\gamma^\star(t)=e^{int}$ in $\mathbb{C}\setminus\{0\}$. Just define$$\begin{array}{rccc}H\colon&[0,2\pi]\times[0,1]&\longrightarrow&\mathbb{C}\setminus\{0\}\\&(t,u)&\mapsto&e^{int}+rue^{imt}.\end{array}$$Clearly, $H(t,0)=\gamma^\star(t)$, whereas $H(t,1)=\gamma(t)$. So, the winding number is $n$.
  Now, suppose that $r>1$. The the loop $\gamma$ is homotopic to the loop $\gamma^\star\colon[0,2\pi]\longrightarrow\mathbb C$ defined by $\gamma^\star(t)=e^{imt}$ in $\mathbb{C}\setminus\{0\}$. The argument is similar. This time you shrink that $e^{int}$ part.
  • The first argument proves that the answer is $n$ when $r\in(0,1)$, whereas the second one proves that the answer is $m$ when $r\in(1,+\infty)$. I don't see how the first argument could prove both cases. – José Carlos Santos Jun 27 '18 at 15:04
explanation: The document contains a math question about calculating the winding number of a given complex function. It also includes an attempt to solve the problem and a detailed solution provided by another user. The solution explains the homotopy argument for two cases: \(0 < r < 1\) and \(r > 1\).
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Winding number of a radius dependent function?
  
  The question: Let $\gamma:[0,2\pi] \rightarrow \mathbb{C}, t \mapsto e^{int} + re^{imt}$, with $n,m \in \mathbb{Z}$ and $0<r \neq 1$. Calculate the winding number.
  
  My attempt: My first try was to calculate the integral $\operatorname{Ind}(\gamma,z) = \int_0^{2\pi} \frac{\gamma'(t)}{\gamma(t) - z}\,\mathrm dz$, but when I solved this integral by subsituting the denominator after some calculation I end up with a logarithm dependent on $r$ and $z$ divided by $2\pi i$ which is obviously wrong since the winding number should be an integer. I suppose there should be a solution by making a homotopy kind of argument without having to calculate the integral but I'm kinda stuck in here.
  
  Greetings a Peaceful Slosh
  
  Suppose that $0<r<1$. Then the loop $\gamma$ is homotopic to the loop $\gamma^\star\colon[0,2\pi]\longrightarrow\mathbb C$ defined by $\gamma^\star(t)=e^{int}$ in $\mathbb{C}\setminus\{0\}$. Just define$$\begin{array}{rccc}H\colon&[0,2\pi]\times[0,1]&\longrightarrow&\mathbb{C}\setminus\{0\}\\&(t,u)&\mapsto&e^{int}+rue^{imt}.\end{array}$$Clearly, $H(t,0)=\gamma^\star(t)$, whereas $H(t,1)=\gamma(t)$. So, the winding number is $n$.
  Now, suppose that $r>1$. The the loop $\gamma$ is homotopic to the loop $\gamma^\star\colon[0,2\pi]\longrightarrow\mathbb C$ defined by $\gamma^\star(t)=e^{imt}$ in $\mathbb{C}\setminus\{0\}$. The argument is similar. This time you shrink that $e^{int}$ part.
  • The first argument proves that the answer is $n$ when $r\in(0,1)$, whereas the second one proves that the answer is $m$ when $r\in(1,+\infty)$. I don't see how the first argument could prove both cases. – José Carlos Santos Jun 27 '18 at 15:04
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document contains a math question about calculating the winding number of a given complex function. It also includes an attempt to solve the problem and a detailed solution provided by another user. The solution explains the homotopy argument for two cases: \(0 < r < 1\) and \(r > 1\).
  
  Label: question-and-answer