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Symbolic computation of a series expansion | Cauchy's theorem

Let us assume we do not know the series expansion at $0$ of $$S : z\mapsto\frac{1}{1-z^2}$$ (consider a more difficult rational fraction if you do not want to make this assumption).

A method to compute the coefficients (*) is to compute a complex integral on a circle: the $k$-th coefficient is given by $$ a(k)=\frac{1}{2i\pi}\int_{C(0,1)}\frac{S(z)}{z^{k+1}}dz=\int_0^1S\left(\exp\left(2i\pi t\right)\right)\exp\left(-2i\pi kt\right) dt. $$

But, even in this easy case, I am not able to help Sagemath to do the computation:

kk,k,t,z=var('kk,k,t,z')
assume(k,'integer')
assume(k>0)
S(z)=1/(1-z^2)
J(kk)=integrate(S(e^(2*I*pi*t))*e^(-2*pi*I*kk*t),t,0,1,hold=True)

Appealing J(k) returns nothing interesting (whether "hold=True" is added or removed) and appealing J(7) (just for test because, indeed a procedure for specific values is not what I am looking for) complains that the integral is divergent... (which is not true).

Any idea to perform the computation of this integral for any integer $k$ (I would like to have a symbolic answer in term of $k$, not a procedure that returns a numerical result for any numeric value of $k$) ? 2.Is there another way to get the coefficients (again meaning a general formula depending on k, like 1/k! for exp).

(*) Not the only one, one also could you a general result on the form of a sequence whose generating function is a rational fraction.

Symbolic computation of a series expansion | Cauchy's theorem

Let us assume we do not know the series expansion at $0$ of $$S : z\mapsto\frac{1}{1-z^2}$$ (consider a more difficult rational fraction if you do not want to make this assumption).

A method to compute the coefficients (*) is to compute a complex integral on a circle: the $k$-th coefficient is given by $$ a(k)=\frac{1}{2i\pi}\int_{C(0,1)}\frac{S(z)}{z^{k+1}}dz=\int_0^1S\left(\exp\left(2i\pi t\right)\right)\exp\left(-2i\pi kt\right) dt. $$

But, even in this easy case, I am not able to help Sagemath to do the computation:

kk,k,t,z=var('kk,k,t,z')
assume(k,'integer')
assume(k>0)
S(z)=1/(1-z^2)
J(kk)=integrate(S(e^(2*I*pi*t))*e^(-2*pi*I*kk*t),t,0,1,hold=True)

Appealing J(k) returns nothing interesting (whether "hold=True" is added or removed) and appealing J(7) (just for test because, indeed a procedure for specific values is not what I am looking for) complains that the integral is divergent... (which is not true).

Any idea to perform the computation of this integral for any integer $k$ (I would like to have a symbolic answer in term of $k$, not a procedure that returns a numerical result for any numeric value of $k$) ? 2.Is there another way to get the coefficients (again meaning a general formula depending on k, like 1/k! for exp).

(*) Not the only one, one also could you a general result on the form of a sequence whose generating function is a rational fraction.

NB. A similar question was asked a few years ago, without positive answer. But situation might have evolved.

Symbolic computation of a series expansion | Cauchy's theorem

Let us assume we do not know the series expansion at $0$ of $$S : z\mapsto\frac{1}{1-z^2}$$ (consider a more difficult rational fraction if you do not want to make this assumption).

A method to compute the coefficients (*) is to compute a complex integral on a circle: the $k$-th coefficient is given by $$ a(k)=\frac{1}{2i\pi}\int_{C(0,1)}\frac{S(z)}{z^{k+1}}dz=\int_0^1S\left(\exp\left(2i\pi t\right)\right)\exp\left(-2i\pi kt\right) dt. $$

But, even in this easy case, I am not able to help Sagemath to do the computation:

kk,k,t,z=var('kk,k,t,z')
assume(k,'integer')
assume(k>0)
S(z)=1/(1-z^2)
J(kk)=integrate(S(e^(2*I*pi*t))*e^(-2*pi*I*kk*t),t,0,1,hold=True)

Appealing J(k) returns nothing interesting (whether "hold=True" is added or removed) and appealing J(7) (just for test because, indeed a procedure for specific values is not what I am looking for) complains that the integral is divergent... (which is not true).

Any idea to perform the computation of this integral for any integer $k$ (I would like to have a symbolic answer in term of $k$, not a procedure that returns a numerical result for any numeric value of $k$) ? 2.Is there another way to get the coefficients (again meaning a general formula depending on k, like 1/k! for exp).

(*) Not the only one, one also could you a general result on the form of a sequence whose generating function is a rational fraction.

NB. A similar question was asked a few years ago, without positive answer. But situation might have evolved.

The full text Sage message :

RuntimeError Traceback (most recent call last) File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/interfaces/maxima_lib.py:817, in MaximaLib.sr_integral(self, *args) 816 try: --> 817 return max_to_sr(maxima_eval(([max_integrate], 818 [sr_to_max(SR(a)) for a in args]))) 819 except RuntimeError as error:

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/libs/ecl.pyx:838, in sage.libs.ecl.EclObject.__call__ (build/cythonized/sage/libs/ecl.c:11700)() 837 lispargs = EclObject(list(args)) --> 838 return ecl_wrap(ecl_safe_apply(self.obj, (<eclobject>lispargs).obj)) 839

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/libs/ecl.pyx:358, in sage.libs.ecl.ecl_safe_apply (build/cythonized/sage/libs/ecl.c:9207)() 357 else: --> 358 raise RuntimeError("ECL says: {}".format(message)) 359 else:

RuntimeError: ECL says: Maxima asks: Is %i*%pi an integer?

During handling of the above exception, another exception occurred:

ValueError Traceback (most recent call last) Cell In[4], line 1 ----> 1 J(k)

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/expression.pyx:6187, in sage.symbolic.expression.Expression.__call__ (build/cythonized/sage/symbolic/expression.cpp:78521)() 6185 z^2 + x^y 6186 """ -> 6187 return self._parent._call_element_(self, args, *kwds) 6188 6189 def variables(self):

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/callable.py:422, in CallableSymbolicExpressionRing_class._call_element_(self, _the_element, *args, kwds) 420 d = dict(zip([repr(_) for _ in self.arguments()], args)) 421 d.update(kwds) --> 422 return SR(_the_element.substitute(d))

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/structure/element.pyx:849, in sage.structure.element.Element.substitute (build/cythonized/sage/structure/element.c:15440)() 847 5 848 """ --> 849 return self.subs(args, *kwds) 850 851

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/expression.pyx:5899, in sage.symbolic.expression.Expression.subs (build/cythonized/sage/symbolic/expression.cpp:76733)() 5897 smap.insert(make_pair((<expression>self.coerce_in(k))._gobj, 5898 (<expression>self.coerce_in(v))._gobj)) -> 5899 res = self._gobj.subs_map(smap, 0) 5900 return new_Expression_from_GEx(self._parent, res) 5901

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/function.pyx:1137, in sage.symbolic.function.BuiltinFunction._evalf_or_eval_ (build/cythonized/sage/symbolic/function.c:16410)() 1135 res = self._evalf_try_(args) 1136 if res is None: -> 1137 return self._eval0_(args) 1138 else: 1139 return res

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/integration/integral.py:254, in DefiniteIntegral._eval_(self, f, x, a, b) 252 for integrator in self.integrators: 253 try: --> 254 A = integrator(*args) 255 except (NotImplementedError, TypeError, 256 AttributeError, RuntimeError): 257 pass

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/symbolic/integration/external.py:46, in maxima_integrator(expression, v, a, b) 44 result = maxima.sr_integral(expression, v) 45 else: ---> 46 result = maxima.sr_integral(expression, v, a, b) 47 return result._sage_()

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/interfaces/maxima_lib.py:827, in MaximaLib.sr_integral(self, *args) 825 raise ValueError("Integral is divergent.") 826 elif "Is" in s: # Maxima asked for a condition --> 827 self._missing_assumption(s) 828 else: 829 raise

File /private/var/tmp/sage-10.3-current/local/var/lib/sage/venv-python3.11.1/lib/python3.11/site-packages/sage/interfaces/maxima_lib.py:1074, in MaximaLib._missing_assumption(self, errstr) 1071 outstr = "Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation may help (example of legal syntax is 'assume("\ 1072 + errstr[jj + 1:k] + ">0)', see assume? for more details)\n" + errstr 1073 outstr = outstr.replace('_SAGE_VAR_', '') -> 1074 raise ValueError(outstr)

ValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation may help (example of legal syntax is 'assume(%i%pi>0)', see assume? for more details) Is %i%pi an integer?

Symbolic computation of a series expansion | Cauchy's theorem

Let us assume we do not know the series expansion at $0$ of $$S : z\mapsto\frac{1}{1-z^2}$$ (consider a more difficult rational fraction if you do not want to make this assumption).

A method to compute the coefficients (*) is to compute a complex integral on a circle: the $k$-th coefficient is given by $$ a(k)=\frac{1}{2i\pi}\int_{C(0,1)}\frac{S(z)}{z^{k+1}}dz=\int_0^1S\left(\exp\left(2i\pi t\right)\right)\exp\left(-2i\pi kt\right) dt. $$

But, even in this easy case, I am not able to help Sagemath to do the computation:

kk,k,t,z=var('kk,k,t,z')
assume(k,'integer')
assume(k>0)
S(z)=1/(1-z^2)
J(kk)=integrate(S(e^(2*I*pi*t))*e^(-2*pi*I*kk*t),t,0,1,hold=True)

Appealing J(k) returns ~~nothing interesting (whether "hold=True"~~ a mistake and requires an assumption on $I$ and $\pi$ :

ValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation may help (example of legal syntax is ~~added or removed) and appealing J(7) (just~~ 'assume(%i%pi>0)', see assume? for ~~test because, indeed a procedure for specific values is not what I am looking for) complains that the integral is divergent... (which is not true).~~more details) Is %i%pi an integer?

Any idea to perform the computation of this integral for any integer $k$ (I would like to have a symbolic answer in term of $k$, not a procedure that returns a numerical result for any numeric value of $k$) ? 2.Is there another way to get the coefficients (again meaning a general formula depending on k, like 1/k! for exp).