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{\displaystyle A_{k}={\frac {2}{2\pi }}\int _{0}^{2\pi }{y\left(\theta \right)sin\left(k\theta \right)\mathrm {d} \theta }}
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{\displaystyle A_{k}={\frac {2}{2\pi }}\left[\int _{\alpha }^{\pi -\alpha }{Y_{M}sin\left(k\theta \right)\mathrm {d} \theta }+\int _{\pi +\alpha }^{2\pi -\alpha }{-Y_{M}sin\left(k\theta \right)\mathrm {d} \theta }\right]}
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{\displaystyle A_{k}={\frac {2Y_{M}}{2\pi }}\left\{\left[{\frac {-cos\left(k\theta \right)}{k}}\right]_{\alpha }^{\pi -\alpha }+\left[{\frac {cos\left(k\theta \right)}{k}}\right]_{\alpha +\pi }^{2\pi -\alpha }\right\}}
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{\displaystyle A_{k}={\frac {Y_{M}}{k\pi }}\left[-cos\left(k\pi -k\alpha \right)+cos\left(k\alpha \right)+cos\left(2k\pi -k\alpha \right)-cos\left(k\pi +k\alpha \right)\right]}
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{\displaystyle A_{k}={\frac {Y_{M}}{k\pi }}\left[{\begin{matrix}-cos\left(k\pi \right)cos\left(k\alpha \right)-sin\left(k\pi \right)sin\left(k\alpha \right)\\+cos\left(k\alpha \right)\\+cos\left(2k\pi \right)cos\left(k\alpha \right)+sin\left(2k\pi \right)sin\left(k\alpha \right)\\-cos\left(k\pi \right)cos\left(k\alpha \right)+sin\left(k\pi \right)sin\left(k\alpha \right)\end{matrix}}\right]}
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{\displaystyle A_{k}={\frac {Y_{M}}{k\pi }}\left[-2cos\left(k\pi \right)cos\left(k\alpha \right)+2cos\left(k\alpha \right)\right]}
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{\displaystyle A_{k}={\frac {Y_{M}}{k\pi }}\left[\left(-2cos\left(k\pi \right)+2\right)cos\left(k\alpha \right)\right]}
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{\displaystyle A_{k}={\frac {Y_{M}}{k\pi }}\left[\left(-2\left(-1\right)^{k}+2\right)cos\left(k\alpha \right)\right]}
Pour tout k pair,
A
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{\displaystyle A_{k}}
Pour k impair (k = 2n+1, n >= 0), on a
![{\displaystyle y(t)={\begin{cases}+Y_{m},&{\text{si }}t\in ]0,{\frac {T}{2}}[\\-Y_{m},&{\text{si }}t\in ]{\frac {T}{2}},T[\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24c66eae942911309796e7f30142d23fcfc72c6d)
![{\displaystyle y(t)={\frac {4Y_{m}}{\pi }}\left\{\sin(\omega t)+{\frac {1}{3}}\sin(3\omega t)+\ldots +{\frac {1}{2n+1}}\sin \left[\left(2n+1\right)\omega t\right]+\ldots \right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bbdb21fa8927c58eee3465f8ee29726c72005528)
![{\displaystyle y(t)={\frac {4Y_{m}}{\pi }}\sum _{n=0}^{\infty }{\frac {1}{2n+1}}\sin \left[\left(2n+1\right)\omega t\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6c2da541dcafd7546a8acd3aaa82ebcc623bb9e)
![{\displaystyle A_{k}={\frac {Y_{m}}{\pi }}\left\{\left[{\frac {-\cos(k\theta )}{k}}\right]_{O}^{\pi }+\left[{\frac {\cos(k\theta )}{k}}\right]_{\pi }^{2\pi }\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99256f23f4dd40aeba9792a15c38f918b3a554f9)
![{\displaystyle A_{k}={\frac {Y_{m}}{\pi }}\left\{\left[{\frac {-\cos(k\pi )}{k}}-{\frac {-\cos(0k)}{k}}\right]+\left[{\frac {\cos(2k\pi )}{k}}-{\frac {\cos(k\pi )}{k}}\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c9ddb48d2f5dec0344af0a2dc1209f022e7bbabd)
![{\displaystyle A_{k}={\frac {Y_{m}}{\pi }}\left\{\left[{\frac {-\cos(k\pi )}{k}}-{\frac {-\cos(0)}{k}}\right]+\left[{\frac {\cos(2k\pi )}{k}}-{\frac {\cos(k\pi )}{k}}\right]\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f811dc6a48b0b2060e4b2a83a55931f915d796b)
![{\displaystyle A_{k}={\frac {Y_{m}}{k\pi }}\left[-2\cos(k\pi )+1+\cos(2k\pi )\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee7bdd8f810e7ee99105c1f983010f8aa6d25639)
![{\displaystyle A_{k}={\frac {Y_{m}}{k\pi }}\left[-2\cos(k\pi )+2\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7547689677fa6e903c3023747feb7fe7facf4e)
![{\displaystyle A_{k}={\frac {2Y_{m}}{k\pi }}\left[-\cos(k\pi )+1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0adffc74763e5959f5a28b64ebff6e3a7b057d79)
![{\displaystyle A_{k}={\frac {2Y_{m}}{k\pi }}\left[-\left(-1\right)^{k}+1\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11840aa9aabad1e0f1c22dce2029ce26aed2079b)
![{\displaystyle y(t)={\begin{cases}0,&{\text{si }}t\in ]0,\alpha [\\+Y_{m},&{\text{si }}t\in ]\alpha ,{\frac {T}{2}}-\alpha [\\0,&{\text{si }}t\in ]{\frac {T}{2}}-\alpha ,{\frac {T}{2}}+\alpha [\\-Y_{m},&{\text{si }}t\in ]{\frac {T}{2}}+\alpha ,T-\alpha [\\0,&{\text{si }}t\in ]T-\alpha ,T[\\\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53d401eeec9e104966541de36de62eff6977795)
![{\displaystyle y(t)={\frac {4Y_{m}}{\pi }}\left\{\cos(\alpha )\sin(\omega t)+{\frac {\cos(3\alpha )}{3}}\sin(3\omega t)+\ldots +{\frac {\cos[(2n+1)\alpha ]}{2n+1}}\sin \left[\left(2n+1\right)\omega t\right]+\ldots \right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/432725e159ff14f62f6b82a62de2ca0d194dd709)
![{\displaystyle y(t)={\frac {4Y_{m}}{\pi }}\sum _{n=0}^{\infty }{\frac {\cos[(2n+1)\alpha ]}{2n+1}}\sin \left[\left(2n+1\right)\omega t\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07af0c0a29ab675aa5373d4172010eeda10d0dd7)
![{\displaystyle A_{k}={\frac {2}{2\pi }}\left[\int _{\alpha }^{\pi -\alpha }{Y_{M}sin\left(k\theta \right)\mathrm {d} \theta }+\int _{\pi +\alpha }^{2\pi -\alpha }{-Y_{M}sin\left(k\theta \right)\mathrm {d} \theta }\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0b140df3e9b172abfd4e547ac912ca313207a99d)
![{\displaystyle A_{k}={\frac {2Y_{M}}{2\pi }}\left\{\left[{\frac {-cos\left(k\theta \right)}{k}}\right]_{\alpha }^{\pi -\alpha }+\left[{\frac {cos\left(k\theta \right)}{k}}\right]_{\alpha +\pi }^{2\pi -\alpha }\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/31386d256bb56423d40aa187491c8d1e3d380c9b)
![{\displaystyle A_{k}={\frac {Y_{M}}{k\pi }}\left[-cos\left(k\pi -k\alpha \right)+cos\left(k\alpha \right)+cos\left(2k\pi -k\alpha \right)-cos\left(k\pi +k\alpha \right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/415f1d93d348504173b3e68e7345c06bcce6954c)
![{\displaystyle A_{k}={\frac {Y_{M}}{k\pi }}\left[{\begin{matrix}-cos\left(k\pi \right)cos\left(k\alpha \right)-sin\left(k\pi \right)sin\left(k\alpha \right)\\+cos\left(k\alpha \right)\\+cos\left(2k\pi \right)cos\left(k\alpha \right)+sin\left(2k\pi \right)sin\left(k\alpha \right)\\-cos\left(k\pi \right)cos\left(k\alpha \right)+sin\left(k\pi \right)sin\left(k\alpha \right)\end{matrix}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/919c7b555af381f175679be67a0d970735881c1c)
![{\displaystyle A_{k}={\frac {Y_{M}}{k\pi }}\left[-2cos\left(k\pi \right)cos\left(k\alpha \right)+2cos\left(k\alpha \right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/206325cc0990250e0c963b0ff363fcf0e480525a)
![{\displaystyle A_{k}={\frac {Y_{M}}{k\pi }}\left[\left(-2cos\left(k\pi \right)+2\right)cos\left(k\alpha \right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4bbb28267d807daa17e134577c16888571e6a81a)
![{\displaystyle A_{k}={\frac {Y_{M}}{k\pi }}\left[\left(-2\left(-1\right)^{k}+2\right)cos\left(k\alpha \right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1d2dab0901cca34690e4a2ec144fd6ded0fcee12)
