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  • https://math.libretexts.org/Courses/East_Tennesee_State_University/Book%3A_Differential_Equations_for_Engineers_(Lebl)_Cintron_Copy/7%3A_Power_series_methods/7.3%3A_Singular_Points_and_the_Method_of_Frobenius
    While behavior of ODEs at singular points is more complicated, certain singular points are not especially difficult to solve. Let us look at some examples before giving a general method. We may be luc...While behavior of ODEs at singular points is more complicated, certain singular points are not especially difficult to solve. Let us look at some examples before giving a general method. We may be lucky and obtain a power series solution using the method of the previous section, but in general we may have to try other things.
  • https://math.libretexts.org/Bookshelves/Calculus/Elementary_Calculus_2e_(Corral)/06%3A_Methods_of_Integration/6.01%3A_Integration_by_Parts
    \[\begin{aligned} d(x\,e^{-x}) ~&=~ x\,d(e^{-x}) ~+~ d(x)\,e^{-x}\\ &=~ -x\,e^{-x}\,\dx ~+~ e^{-x}\,\dx\\ d(x\,e^{-x}) ~&=~ -x\,e^{-x}\,\dx ~-~ d(e^{-x})\\ x\,e^{-x}\,\dx ~&=~ -d(x\,e^{-x}) ~-~ d(e^{-...\[\begin{aligned} d(x\,e^{-x}) ~&=~ x\,d(e^{-x}) ~+~ d(x)\,e^{-x}\\ &=~ -x\,e^{-x}\,\dx ~+~ e^{-x}\,\dx\\ d(x\,e^{-x}) ~&=~ -x\,e^{-x}\,\dx ~-~ d(e^{-x})\\ x\,e^{-x}\,\dx ~&=~ -d(x\,e^{-x}) ~-~ d(e^{-x}),\quad\text{so integrate both sides to get}\
  • https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/14%3A_Analytic_Continuation_and_the_Gamma_Function/14.04%3A_Proofs_of_(some)_properties
    L(f;s)=sL(tz;s)=Γ(z+1)sz Γ(z)=Γ(z+1)z=Γ(z+2)(z+1)z \[\Gamma (z) = \dfr...L(f;s)=sL(tz;s)=Γ(z+1)sz Γ(z)=Γ(z+1)z=Γ(z+2)(z+1)z Γ(z)=Γ(z+m+1)(z+m)(z+m1)+ ...+(z+1)z Res(Γ,m)=lim
  • https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/05%3A_Non-sinusoidal_Harmonics_and_Special_Functions/5.04%3A_Gamma_Function
    A function that often occurs in the study of special functions is the Gamma function. We will need the Gamma function in the next section on Fourier-Bessel series.
  • https://math.libretexts.org/Bookshelves/Analysis/Complex_Variables_with_Applications_(Orloff)/14%3A_Analytic_Continuation_and_the_Gamma_Function/14.02%3A_Definition_and_properties_of_the_Gamma_function
    \Gamma (z) = [ze^{\gamma z} \prod_{1}^{\infty} (1 + \dfrac{z}{n}) e^{-z/n}]^{-1}, where \gamma is Euler's constant \Gamma (z + 1) \approx \sqrt{2\pi} z^{z + 1/2} e^{-z} for |z| large, ...\Gamma (z) = [ze^{\gamma z} \prod_{1}^{\infty} (1 + \dfrac{z}{n}) e^{-z/n}]^{-1}, where \gamma is Euler's constant \Gamma (z + 1) \approx \sqrt{2\pi} z^{z + 1/2} e^{-z} for |z| large, \text{Re} (z) > 0. Use the properties of \Gamma to show that \Gamma (1/2) = \sqrt{\pi} and \Gamma (3/2) = \sqrt{\pi}/2. 2^0 \Gamma \left(\dfrac{1}{2}\right) \Gamma (1) = \sqrt{\pi} \Gamma (1) \Rightarrow \Gamma \left(\dfrac{1}{2}\right) = \sqrt{\pi}. \nonumber
  • https://math.libretexts.org/Bookshelves/Differential_Equations/Partial_Differential_Equations_(Walet)/10%3A_Bessel_Functions_and_Two-Dimensional_Problems/10.03%3A_Gamma_Function
    For ν not an integer the recursion relation for the Bessel function generates something very similar to factorials. These quantities are most easily expressed in something called a Gamma-function.
  • https://math.libretexts.org/Bookshelves/Differential_Equations/A_First_Course_in_Differential_Equations_for_Scientists_and_Engineers_(Herman)/04%3A_Series_Solutions/4.07%3A_Gamma_Function
    A FUNCTION THAT OFTEN OCCURS IN THE STUDY OF SPECIAL FUNCTIONS is the Gamma function. We will need the Gamma function in the next section on Fourier-Bessel series.
  • https://math.libretexts.org/Bookshelves/Differential_Equations/Differential_Equations_for_Engineers_(Lebl)/7%3A_Power_series_methods/7.3%3A_Singular_Points_and_the_Method_of_Frobenius
    While behavior of ODEs at singular points is more complicated, certain singular points are not especially difficult to solve. Let us look at some examples before giving a general method. We may be luc...While behavior of ODEs at singular points is more complicated, certain singular points are not especially difficult to solve. Let us look at some examples before giving a general method. We may be lucky and obtain a power series solution using the method of the previous section, but in general we may have to try other things.
  • https://math.libretexts.org/Courses/Mt._San_Jacinto_College/Differential_Equations_(No_Linear_Algebra_Required)/06%3A_Series_Solutions_of_Linear_Equations/6.03%3A_Singular_Points_and_the_Method_of_Frobenius
    \[ \begin{align}\begin{aligned} 0 &= 4x^2y''-4x^2y'+(1-2x)y \\ &= 4x^2 \, \left( \sum_{k=0}^\infty (k+r)\,(k+r-1) \, a_k x^{k+r-2} \right)-4x^2 \, \left( \sum_{k=0}^\infty (k+r) \, a_k x^{k+r-1} \righ...\[ \begin{align}\begin{aligned} 0 &= 4x^2y''-4x^2y'+(1-2x)y \\ &= 4x^2 \, \left( \sum_{k=0}^\infty (k+r)\,(k+r-1) \, a_k x^{k+r-2} \right)-4x^2 \, \left( \sum_{k=0}^\infty (k+r) \, a_k x^{k+r-1} \right)+(1-2x) \left( \sum_{k=0}^\infty a_k x^{k+r} \right) \\ &=\left( \sum_{k=0}^\infty 4 (k+r)\,(k+r-1) \, a_k x^{k+r} \right)-\left( \sum_{k=0}^\infty 4 (k+r) \, a_k x^{k+r+1} \right)+\left( \sum_{k=0}^\infty a_k x^{k+r} \right)-\left( \sum_{k=0}^\infty 2a_k x^{k+r+1} \right) \\ &=\left( \sum_{k=0}^…
  • https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/MAT-204%3A_Differential_Equations_for_Science_(Lebl_and_Trench)/09%3A_Power_series_methods/9.03%3A_Singular_Points_and_the_Method_of_Frobenius
    While behavior of ODEs at singular points is more complicated, certain singular points are not especially difficult to solve. Let us look at some examples before giving a general method. We may be luc...While behavior of ODEs at singular points is more complicated, certain singular points are not especially difficult to solve. Let us look at some examples before giving a general method. We may be lucky and obtain a power series solution using the method of the previous section, but in general we may have to try other things.
  • https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/16%3A_Power_series_methods/16.03%3A_Singular_Points_and_the_Method_of_Frobenius
    While behavior of ODEs at singular points is more complicated, certain singular points are not especially difficult to solve. Let us look at some examples before giving a general method. We may be luc...While behavior of ODEs at singular points is more complicated, certain singular points are not especially difficult to solve. Let us look at some examples before giving a general method. We may be lucky and obtain a power series solution using the method of the previous section, but in general we may have to try other things.

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