From the result of the last section and the equations (1) of § 228 it follows at once that cosz=1z22!+z44!,sinz=zz33!+z55! for all values of z. These results were proved for real values of z in Ex. LVI. 1.

Example XCVI
1. Calculate cosi and sini to two places of decimals by means of the power series for cosz and sinz.

2. Prove that |cosz|cosh|z| and |sinz|sinh|z|.

3. Prove that if |z|<1 then |cosz|<2 and |sinz|<65|z|.

4. Since sin2z=2sinzcosz we have (2z)(2z)33!+(2z)55!=2(zz33!+)(1z22!+). Prove by multiplying the two series on the right-hand side (§ 195) and equating coefficients (§ 194) that (2n+11)+(2n+13)++(2n+12n+1)=22n. Verify the result by means of the binomial theorem. Derive similar identities from the equations cos2z+sin2z=1,cos2z=2cos2z1=12sin2z.

5. Show that exp{(1+i)z}=0212nexp(14nπi)znn!.

6. Expand coszcoshz in powers of z. [We have coszcoshz+isinzsinhz=cos{(1i)z}=12[exp{(1+i)z}+exp{(1+i)z}]=120212n{1+(1)n}exp(14nπi)znn!, and similarly coszcoshzisinzsinhz=cos(1+i)z=120212n{1+(1)n}exp(14nπi)znn!. Hence coszcoshz=120212n{1+(1)n}cos14nπznn!=122z44!+24z88!.]

7. Expand sinzsinhz, coszsinhz, and sinzcoshz in powers of z.

8. Expand sin2z and sin3z in powers of z. [Use the formulae sin2z=12(1cos2z),sin3z=14(3sinzsin3z), . It is clear that the same method may be used to expand cosnz and sinnz, where n is any integer.]

9. Sum the series C=1+cosz1!+cos2z2!+cos3z3!+,S=sinz1!+sin2z2!+sin3z3!+.

[Here C+iS=1+exp(iz)1!+exp(2iz)2!+=exp{exp(iz)}=exp(cosz){cos(sinz)+isin(sinz)}, and similarly CiS=exp{exp(iz)}=exp(cosz){cos(sinz)isin(sinz)}. Hence C=exp(cosz)cos(sinz),S=exp(cosz)sin(sinz).]

10. Sum 1+acosz1!+a2cos2z2!+,asinz1!+a2sin2z2!+.

11. Sum 1cos2z2!+cos4z4!,cosz1!cos3z3!+ and the corresponding series involving sines.

12. Show that 1+cos4z4!+cos8z8!+=12{cos(cosz)cosh(sinz)+cos(sinz)cosh(cosz)}.

13. Show that the expansions of cos(x+h) and sin(x+h) in powers of h (Ex. LVI. 1) are valid for all values of x and h, real or complex.


232. The power series for expz Main Page 234–235. The logarithmic series