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With correct syntax: In[2]:= Integrate[(2 x^2/a) Sin[n Pi x/a]^2, {x, 0, a}] Out[2]= (a^2 (4 n^3 \[Pi]^3 - 6 n \[Pi] Cos[2 n \[Pi]] + (3 - 6 n^2 \[Pi]^2) Sin[ 2 n \[Pi]]))/(12 n^3 \[Pi]^3)
A fast way to get an exact fraction into its real-number equivalent is to multiply by 1.0 (or just 1.) Example: In[1]:= x = 355/113 Out[1]= 355/113 In[2]:= 1.0 x Out[2]= 3.14159 In[3]:= N[x] ...
Thanks Sam! That’s great. Solves the problem without any need for Mathematica code update.
Try this, with x=1/z In[1]:= PadeApproximant[(z - 1)/ z ((K z)/(z - 1) - K/(z - E^(-T/Ts))) /. z -> 1/x, {x, 0, {1, 1}}] Out[1]= (K - E^(-(T/Ts)) K x)/(1 + E^(-(T/Ts)) (-1 + E^(T/Ts)) x) or this In[6]:=...
In simple terms, x = Cos[theta] is a multivalued function. Many values of theta (in fact, an infinite number, theta + 2 n pi) give the same value of x. In mathematical jargon, ArcCos[x] is an injective but not a bijective function.
The asymptotic expansions for modified Bessel functions are inherently complex functions. See asymptotic expansion for I_nu(z) in ?? ?
Construct a 3x3 matrix of random numbers and replace the diagonal elements by {x,y,z}: In[1]:= matrix = Table[Random[], {n, 1, 3}, {m, 1, 3}] Out[1]= {{0.605062, 0.78659, 0.705669}, {0.0363308, 0.416557, 0.259851}, {0.34089,...
Mathematica is evidently not able to do the integral analytically. You need some clever transformation of variables or just a resort to numerical integration (with g, a and s given numerical values).
A simplification of a definite integral would be a numerical result! In[214]:= F = N[Integrate[(x^4*Exp[x])/(Exp[x] - 1)^2, {x, 0, 44}]] Out[214]= 25.9758
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