# Complex Numbers II

• ### Circular/ Trigonometric Functions

The sine and cosine functions are known as circular functions, because ${(cos \ \theta, sin \ \theta)}$ represents a point on the unit circle. Also,

${cos^2 \theta + sin^2 \theta = 1}$

From Euler’s formula, ${e^{i \theta} = cos \ \theta + i \ sin \ \theta}$,

${cos \ \theta = \frac {e^{i \theta} + e^{-i \theta}}{2}, sin \ \theta = \frac {e^{i \theta} - e^{- i \theta}}{2i}}$

These definitions are valid for any number ${\theta}$.

• ### Hyperbolic Functions

The hyperbolic functions parameterize the hyperbola ${x^2-y^2 =1}$ (just like circular functions parameterize the circle). They appear in solutions of many linear differential equations. They are defined as:

${cosh \ x = \frac {e^x + e^{-x}}{2}, sinh \ x = \frac {e^x - e^{-x}}{2}}$

${tanh \ x = \frac {sinh \ x}{cosh \ x}, coth\ x = \frac 1 {tanh \ x}}$

${sech \ x = \frac 1 {cosh \ x}, cosech \ x = \frac 1 {sinh \ x}}$

• ### Relationship between Circular and Hyperbolic Functions

${sin (ix) = i \ sinh (x)}$

${cos (ix) = cosh (x)}$

${tan (ix) = i \ tanh (x)}$

• ### Other Formulas

${cosh^2 x - sinh^2 x =1}$

${sech^2 x = 1 - tanh^2 x}$

${cosech^2 x = 1 - coth^2 x}$

By using the definitions, we can also write down the expressions for derivatives and integrations of hyperbolic functions.

• ### Inverse Hyperbolic Functions

${sinh^{-1}x = log (x + \sqrt {x^2+1})}$

${cosh^{-1}x = log (x + \sqrt {x^2-1})}$

${tanh^{-1}x = \frac 1 2 log \big ( \frac {1+x}{1-x} \big)}$

• ### Important Trigonometric Formulas

Also see these: post 1 and post 2.

${sin (A+B) = sin(A)cos(B) \ + \ cos(A)sin(B)}$

${sin (A-B) = sin(A)cos(B) \ - \ cos(A)sin(B)}$

${cos (A+B) = cos(A)cos(B) \ - \ sin(A)sin(B)}$

${cos (A-B) = cos(A)cos(B) \ + \ sin(A)sin(B)}$

• ### Logarithms of Complex Numbers

We use the polar representation of complex numbers. Let ${z = x+iy = re^{i \theta}}$

${log (z) = log (r e^{i \theta}) = log (r) + i \theta = log (r) + i (2n \pi + tan^{-1} \frac y x)}$

Thus, ${log}$ of a complex number is multivalued. The principal value of log is

${Log(z) = log (r)+ i tan^{-1} \frac y x}$

• ### Types of Problems on Logarithms of Complex Numbers

The problems ask for separating real and imaginary parts of a complicated expression.

I) ${(x+iy)^{a+ib}}$ : Use ${(x+iy)^{a+ib} = e^{(a+ib) Log (x+iy)}}$

II) ${Log_{(a+ib)} (x+iy)}$ : Use change of base rule.

III) ${\frac {(a+ib)^{x+iy}}{(p+iq)^{r+is}} = c+id}$ : Take ${Log}$ on both sides.