An Elementary Treatise On Calculus
An Elementary Treatise On Calculus
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'x=0 ■1 /»a;=Zn- = - I y sinnx • dx (3) -1 nr=2w = — I y cosnx ' dx (4) ^ Jx=0 A complete proof of Fourier's theorem would involve a proof that the right-hand member of equation (1) is a convergent series. * This, however, will here be taken for granted. The proof then reduces to a derivation of equations (2), (3) and (4).
* This matter is discussed in Chapter III of Byerly's Fourier^ s Series and Spherical Harmonics, Ginn and Co. , Boston, 1893.
200 CALCULUS.
In this proof a consideration of a...verage values is most important, and the student should therefore look over Art. 73 again.
Another matter of import- ance is to understand clearly 'V^A^Bin nx ^ ^^le meaning of the equation y = sin nx (and y = cos nx) when n is an integer. Now y = sin nx is the equation of a sine curve. If n = 1 this sine curve makes a posi- tive "arch" between x = and x = TT, and a negative "arch'* between x = tt and x = 27r.
In general the sine curve y = sin nx makes a positive arch between x = and x = -, and a negative arch between x = - n' n and X = — as shown in Fig.
What to read after An Elementary Treatise On Calculus?
You can find similar books in the "Read Also" column, or choose other free books by William S William Suddards Franklin to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
'x=0 ■1 /»a;=Zn- = - I y sinnx • dx (3) -1 nr=2w = — I y cosnx ' dx (4) ^ Jx=0 A complete proof of Fourier's theorem would involve a proof that the right-hand member of equation (1) is a convergent series. * This, however, will here be taken for granted. The proof then reduces to a derivation of equations (2), (3) and (4).
* This matter is discussed in Chapter III of Byerly's Fourier^ s Series and Spherical Harmonics, Ginn and Co. , Boston, 1893.
200 CALCULUS.
In this proof a consideration of a...verage values is most important, and the student should therefore look over Art. 73 again.
Another matter of import- ance is to understand clearly 'V^A^Bin nx ^ ^^le meaning of the equation y = sin nx (and y = cos nx) when n is an integer. Now y = sin nx is the equation of a sine curve. If n = 1 this sine curve makes a posi- tive "arch" between x = and x = TT, and a negative "arch'* between x = tt and x = 27r.
In general the sine curve y = sin nx makes a positive arch between x = and x = -, and a negative arch between x = - n' n and X = — as shown in Fig.
What to read after An Elementary Treatise On Calculus?
You can find similar books in the "Read Also" column, or choose other free books by William S William Suddards Franklin to read onlineMoreLess
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