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.

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