Derivatives for any number $z = x+iy$
Identities for any number $z = x+iy$
Zeros for any number $z = x+iy$
Introduction to Hyperbolic Trig Functions
Section 17.7: 2, 3, 4, 12, 14, 15, 19, 21, 23, 27, 28, 31, 32.
$$ \cos \theta = \frac{e^{i\theta} + e^{-i\theta}}{2} = \cos(x) \cosh(y) - i \cdot \sin(x) \sinh(y) \\
\sin \theta = \frac{e^{i\theta} - e^{-i\theta}}{2i} = \sin(x) \cosh(y) + i \cdot \cos(x) \sinh(y) \\ $$
Derivatives for any number $z = x+iy$
Identities for any number $z = x+iy$
Zeros for any number $z = x+iy$
$$ \cosh(z) = \frac{e^z + e^{-z}}{2} = \cosh(x) \cos(y) - i \cdot \sinh(x) \sin(y) \\
\sinh(z) = \frac{e^z - e^{-z}}{2} = \sinh(x) \cos(y) + i \cdot \cosh(x) \sin(y) $$
Hyperbolic Sine and Cosine for any number $z = x+iy$
Zeros
Section 1.2: 25, 28, 31, 32.
Solve $\frac{dy}{dx} = f(x,y)$ subject to $y(x_0) = y_0$
Solve $\frac{d^2y}{dx^2} = f(x,y,y’)$ subject to $y(x_0) = y_0, y’(x_0) = y_1$
Direction fields
Autonomous First-Order DEs: $\frac{dy}{dx} = f(y)$