From the result of the last section and the equations (1) of § 228 it follows at once that \[\cos z = 1 – \frac{z^{2}}{2!} + \frac{z^{4}}{4!} – \dots,\quad \sin z = z – \frac{z^{3}}{3!} + \frac{z^{5}}{5!} – \dots\] for all values of \(z\). These results were proved for real values of \(z\) in Ex. LVI. 1.

Example XCVI
1. Calculate \(\cos i\) and \(\sin i\) to two places of decimals by means of the power series for \(\cos z\) and \(\sin z\).

2. Prove that \(|\cos z| \leq \cosh|z|\) and \(|\sin z| \leq \sinh|z|\).

3. Prove that if \(|z| < 1\) then \(|\cos z| < 2\) and \(|\sin z| < \frac{6}{5}|z|\).

4. Since \(\sin 2z = 2\sin z \cos z\) we have \[(2z) – \frac{(2z)^{3}}{3!} + \frac{(2z)^{5}}{5!} – \dots = 2\left(z – \frac{z^{3}}{3!} + \dots\right) \left(1 – \frac{z^{2}}{2!} + \dots\right).\] Prove by multiplying the two series on the right-hand side (§ 195) and equating coefficients (§ 194) that \[\binom{2n + 1}{1} + \binom{2n + 1}{3} + \dots + \binom{2n + 1}{2n + 1} = 2^{2n}.\] Verify the result by means of the binomial theorem. Derive similar identities from the equations \[\cos^{2}z + \sin^{2}z = 1,\quad \cos2z = 2\cos^{2}z – 1 = 1 – 2\sin^{2}z.\]

5. Show that \[\exp\{(1 + i)z\} = \sum_{0}^{\infty} 2^{\frac{1}{2}n} \exp(\tfrac{1}{4}n\pi i) \frac{z^{n}}{n!}.\]

6. Expand \(\cos z \cosh z\) in powers of \(z\). [We have \[\begin{aligned} \cos z \cosh z + i\sin z \sinh z &= \cos\{(1 – i)z\} = \tfrac{1}{2} [\exp\{(1 + i)z\} + \exp\{-(1 + i)z\}]\\ &= \tfrac{1}{2} \sum_{0}^{\infty} 2^{\frac{1}{2}n} \{1 + (-1)^{n}\} \exp(\tfrac{1}{4}n\pi i) \frac{z^{n}}{n!},\end{aligned}\] and similarly \[\cos z \cosh z – i\sin z \sinh z = \cos (1 + i)z = \tfrac{1}{2} \sum_{0}^{\infty} 2^{\frac{1}{2}n} \{1 + (-1)^{n}\} \exp(-\tfrac{1}{4}n\pi i) \frac{z^{n}}{n!}.\] Hence \[\cos z \cosh z = \tfrac{1}{2} \sum_{0}^{\infty} 2^{\frac{1}{2}n}\{1 + (-1)^{n}\} \cos \tfrac{1}{4}n\pi \frac{z^{n}}{n!} = 1 – \frac{2^{2}z^{4}}{4!} + \frac{2^{4}z^{8}}{8!} – \dots.]\]

7. Expand \(\sin z \sinh z\), \(\cos z \sinh z\), and \(\sin z \cosh z\) in powers of \(z\).

8. Expand \(\sin^{2} z\) and \(\sin^{3} z\) in powers of \(z\). [Use the formulae \[\sin^{2} z = \tfrac{1}{2} (1 – \cos 2z),\quad \sin^{3} z = \tfrac{1}{4} (3\sin z – \sin 3z),\ \dots.\] It is clear that the same method may be used to expand \(\cos^{n} z\) and \(\sin^{n} z\), where \(n\) is any integer.]

9. Sum the series \[C = 1 + \frac{\cos z}{1!} + \frac{\cos 2z}{2!} + \frac{\cos 3z}{3!} +\dots,\quad S = \frac{\sin z}{1!} + \frac{\sin 2z}{2!} + \frac{\sin 3z}{3!} + \dots.\]

[Here \[\begin{aligned} C + iS &= 1 + \dfrac{\exp(iz)}{1!} + \dfrac{\exp(2iz)}{2!} + \dots = \exp\{\exp(iz)\} \\ &= \exp(\cos z) \{\cos(\sin z) + i\sin(\sin z)\},\end{aligned}\] and similarly \[C – iS = \exp\{\exp(-iz)\} = \exp(\cos z)\{\cos(\sin z) – i\sin(\sin z)\}.\] Hence \[C = \exp(\cos z)\cos(\sin z),\quad S = \exp(\cos z)\sin(\sin z).]\]

10. Sum \[1 + \frac{a\cos z}{1!} + \frac{a^{2}\cos 2z}{2!} + \dots,\quad \frac{a\sin z}{1!} + \frac{a^{2}\sin 2z}{2!} + \dots.\]

11. Sum \[1 – \frac{\cos 2z}{2!} + \frac{\cos 4z}{4!} – \dots,\quad \frac{\cos z}{1!} – \frac{\cos 3z}{3!} + \dots\] and the corresponding series involving sines.

12. Show that \[1 + \frac{\cos 4z}{4!} + \frac{\cos 8z}{8!} + \dots = \tfrac{1}{2}\{\cos(\cos z) \cosh(\sin z) + \cos(\sin z) \cosh(\cos z)\}.\]

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.


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