3.1.2.1 Variable

In the mathematical sense, a variable is a placeholder for a value we do not know at the moment. Usually, it is desirable to write relations not using numbers. In many cases, these numbers change and we do not want to redo the whole calculation. Therefore, we use placeholders, or variables.

3.1.2.2 Equations

If we sum up several variables and establish correlations and relationships between them, we end up with something that is generally considered an equation. An equation puts several variables in a context that is physically meaningful. As an example, when calculating the weight of an object for which we know the density ρ and the volume V , we usually write

$m=\rho V$

Eq. 3.1 is an equation. It puts several variables in a physically meaningful context. Equations are valid for all values we plug into the variables. Obviously, we may need to adapt the units if they are not provided in International System of Units (SI) format, but other than that, this equation will always hold true. We can even reformulate this equation if we know the volume V and the mass m of an object, but not the density. In this case, Eq. 3.1 becomes

$\rho =\frac{m}{V}$

3.1.2.3 Functions

A function is a special type of an equation. In a function, we select one variable that we can choose at will, which is referred to as the independent variable. Depending on the value of this independent variable, other values of an equation have to change. These values are dependent on the values of the independent variable and are therefore referred to as the dependent variable. Returning to our example, Eq. 3.1, we can say that the density ρ and the volume V can be changed as desired. In this case, our problem has two independent variables (which is common). Depending on the values of these two variables, the third variable, the mass (m), has to change in order to balance the equation. Therefore, the mass, m, is the dependent variable in our problem.

Whenever we find a dependency that describes how a dependent variable changes when the independent variable changes, we have what we refer to as a function. A function is therefore a “recipe” that tells us how to interlink the independent variable with the dependent variable. Please note that a function can only have one dependent variable but (theoretically) an unlimited number of independent variables.

Usually, we indicate functions by writing the independent variables in parentheses. Therefore, we could rewrite Eq. 3.1 as

$m\left(\rho ,V\right)=\rho V$

which would be a valid function definition. We could imagine a scenario where we want to determine the mass of different fluids but always take a given volume V₀ in each experiment. Therefore, we could consider the volume to be a constant rather than an independent variable. In this case, we would rewrite Eq. 3.3 as

$m\left(\rho \right)=\rho V_0$

where we have a function with only one independent variable. Please note that whenever the independent variables have been defined (or can be derived from context) functions are often written with the independent variables in parentheses. This is sometimes misleading but common.

3.1.3.1 A Second Example: The Boiled Egg

Another very simple example of a differential equation is the following. Imagine you cook yourself a boiled egg. After the egg has been in the boiling water for a given amount of time, you take the egg out of the boiling water and drop it into cold water. The question would be: What is the temperature of the egg in the cold water as a function of time? At the start of the experiment, the egg has a given initial temperature T₀, which should be somewhere around 100 C given that you just boiled it in water at about this temperature.

There is a certain intuition to this problem. We could assume that heat will flow from the egg into the cold water at a given rate. The more heat it loses, the colder the egg will become. We would also assume that the amount of heat flowing per unit of time is proportional to the difference in temperature between the egg and its surrounding area. Therefore the change of temperature in the egg will also be proportional to this difference, or in other words,

$\begin{array}{l}-\frac{dT}{dt}\infty T\hfill \\ \frac{dT}{dt}=-cT\hfill \end{array}$

This is a simple example of a differential equation that can be solved quickly. In this case, it can be solved by a separation of variables and partial integration (see section 8.2.2), resulting in

$\begin{array}{l}\frac{dT}{dt}=-cT\\ \frac{dT}{T}=-cdt\\ 1\text{n}T=-ct+c_0\\ T=c_{\text{1}}{e}^{-ct}\end{array}$

As you can see, we have an integration constant which is given by the initial value: T (t = 0) = c₁ = T₀. Therefore, we see that the temperature of the egg will decrease exponentially over time.

This is a very good example of a very practical example of a differential equation.

3.1.3.2 Degree of a Differential Equation

In many differential equations we may come across higher derivatives of the function. The degree n of a differential equation is defined by the highest derivative used. The following list gives a few examples:

$\begin{array}{l}\frac{df}{dx}=0\text{first-order}\text{ODE}\\ \frac{\partial f}{\partial x}=0\text{first-order}\text{PDE}\\ \frac{d^{2}f}{d{x}^2}+\frac{d^{2}f}{d{x}^2}=0\text{second-order}\text{ODE}\\ \frac{{\partial }^{2}f}{\partial x^2}+\frac{\partial f}{\partial x}=0\text{second-order}\text{PDE}\end{array}$

3.1.3.3 Homogeneous and Inhomogeneous Differential Equations

A differential equation is considered to be homogeneous if, after moving all terms containing the function f to the left-hand side of the equation, the right-hand side of the equation is 0. If the right-hand side of the equation is nonzero, the equation is considered to be inhomogeneous. It is of no importance if the right-hand side term is a simple constant or a function. The following list expands on this example:

$\begin{array}{lll}\frac{{\partial }^{2}f}{\partial x^2}+\frac{\partial f}{\partial x}\hfill & =0\hfill & \text{second-order homogeneous PDE}\hfill \\ \frac{{\partial }^{2}f}{\partial x^2}+\frac{\partial f}{\partial x}\hfill & =2\hfill & \text{second-order homogeneous PDE}\hfill \\ \frac{{\partial }^{2}f}{\partial x^2}+\frac{\partial f}{\partial x}\hfill & =x^2+y^2\hfill & \text{second-order inhomogeneous PDE}\hfill \\ \frac{d^{2}f}{d{x}^2}+\frac{d^{2}f}{d{x}^2}\hfill & =0\hfill & \text{second-order homogeneous ODE}\hfill \\ \frac{\partial f}{\partial x}\hfill & =0\hfill & \text{first-order homogeneous PDE}\hfill \\ \frac{d^{2}f}{d{x}^2}\hfill & =\frac{dp}{dz}\hfill & \text{first-order inhomogeneous ODE}\hfill \end{array}$

Please note the last example as being an interesting one. Here, the right-hand side is not only an equation, but it is also a differential equation. However, this differential equation depends on z and not on the independent variable x of our differential equation. Thus the differential equation can be solved assuming $\frac{dp}{dx}$ to be a constant.

3.1.3.4 Obtaining the Differential Equation

Often, obtaining the differential equation may be difficult. The differential equation contains the underlying physics. Therefore, a sound understanding of the physical phenomena associated with the problem is necessary; otherwise, the differential equation cannot be obtained. Interestingly, in most cases obtaining the differential equation is not the most problematic point.

3.1.3.5 Solving the Differential Equation

In most cases, the differential equation can be obtained readily, but actually solving it may be very difficult. This is where the beauty and the irony come together: Differential equations often describe the most complex phenomena in a short, precise, and accurate form. However, in many cases obtaining the differential equation is as far as we get to solving the problem. A good example that is highly relevant for fluid mechanics is the Navier-Stokes equation, Eq. 11.40. Everything we need in order to describe even the most complex fluid mechanical effects is contained in this differential equation. But solving this equation is, in most cases, simply impossible. Luckily, many technically relevant, specialized cases exist in which solutions to this equation can actually be found or, at least, approximated.

3.1.4.1 Ordinary Differential Equations

Ordinary differential equations are differential equations of functions that have only one independent variable. Eq. 3.6, which we just discussed, is an example of an ODE. In our example, the volume of the lake is a function only of the time $\frac{d{V}_{\text{lake}}}{dt}$. Thus the change of the function as we change the independent variable is the only change we need to consider. In such a case, the change of the function can be written using the notation $\frac{df}{dx}$. This denotes an ordinary or total differential, indicating that there are no other independent variables that need to be taken into account. In our example, this was the change of the lake’s volume with respect to time $\frac{d{V}_{\text{lake}}}{dt}$.

We will not discuss the rules of forming derivatives here because they can be looked up in any good differential calculus textbook.

3.1.4.2 Partial Differential Equations

If a function has more than one independent variable, the total change in the function will depend on more than one variable. In such a case, we are faced with a “partial differential equation” (PDE). Eq. 3.3 was an example of a function that depends on more than one independent variable. If we want to determine how a change in the volume V influences the mass m (which is the function we are investigating) we need to treat the function as if the second independent variable, the density ρ, was a constant. In this case, we use the notation $\frac{\partial m}{\partial V}$, that indicates that (for the given case) we only want to consider the change of the mass as we change the volume, keeping all the other independent variables constant.

These special types of derivatives are referred to as partial derivatives, which give rise to the term PDE. They are formed exactly as such: Simply keep all other independent variables constant and form a regular derivative. As an example, we will quickly form the two partial derivatives of Eq. 3.3, which are

$\begin{array}{l}\frac{\partial m}{\partial \rho }=\frac{\partial \left(V\rho \right)}{\partial \rho }=V\frac{d\rho }{d\rho }=V\\ \frac{\partial m}{\partial V}=\frac{\partial \left(\rho V\right)}{\partial V}=\rho \frac{dV}{dV}=\rho \end{array}$

As you can see, in both cases the second independent variable is simply treated as a constant that allows forming the ordinary derivative of the second independent variable.

3.1.4.3 ODEs and PDEs

If a differential equation contains a partial derivative it is referred to as a PDE. If it only contains regular derivatives it is referred to as an ODE. In general, ODEs are significantly simpler to solve than PDEs. Unfortunately, PDEs are more common in practical physics and we therefore need to understand how to solve these equations.

3.1.4.4 Comment on the Usage of Independent Variable Lists

In section 3.1.2.3 we have already briefly stated that the independent variables are often not appended to the function, especially if they have been clearly defined. You will often find that whenever a function is written as a differential there is no need to state the independent variable. If a differential is written as $\frac{df}{dx}$ one can clearly see that the dependent variable (or function) f has only one independent variable, which is x. If f had more than one independent variable we must write $\frac{df}{dx}$. In this case, we can tell that f has more than one independent variable and x is one of the independent variables. In this case, it is advisable to repeat the independent variable list and write $\frac{\partial f\left(x,y\right)}{\partial x}$. However, in many cases we usually do not look for the partial differential of only one a variable but for the effect of all variables. Therefore we may come across an equation of the form

$\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}=0$

Here we would assume that the two independent variables of f are x and y, and only those two. In this case, it is acceptable not to repeat them again. You will find these rules being applied very inconsistently in the literature. Most authors argue that it keeps equation expressions shorter and more readable if the dependent variable list is not repeated every time. If the problem has been clearly introduced and the reader is aware of the independent variable, this seems acceptable in many cases. However, if a differential equation is only briefly mentioned and the reader has no notion about the meaning of the individual terms and from where they have been derived, it is often advisable to repeat the independent variables.

3.1.5.1 Total Differential

If f is only dependent on one variable, only one slope must be taken into account and the differential is referred to as the total differential. As stated, it is denoted $\frac{df}{dx}$ in case of the differential of f (x) with respect to x. The total differential represents the overall change df if the dependent variable x changes by dx.

3.1.5.2 Total Differential From Partial Differentials

As we already stated, if a function f (x, y) is dependent on two independent variables x and y, we need to apply partial derivatives. However, we are still interested in the total differential of f (x, y), which has to be expressed in terms of changes with respect to both of the independent variables x and y. Surprisingly it is very easy to study this case graphically (see Fig. 3.2b). Following up on the concept of a slope along a changing variable, this two-dimensional case gives rise to two tangential planes (in three-dimensional space). The first plane will be defined by the slope of the function along x, and the second plane will be defined by the slope of the function along y. Both slopes are defined at the point S₀. Each of these two slopes will contribute to the change of the function’s value df and the total differential is defined as

$df=d{f}_x+d{f}_y$

Following Eq. 3.8 the two values df_x (originating from the tangential plane along the x-axis) and df_x (originating from the tangential plane along the y-axis) can be calculated from the derivatives of f (x, y) along x and y in S₀, respectively. As discussed, these two partial derivatives are calculated with respect to a given axis while ignoring any change along the other axis or rather while keeping the value for the other axis fixed. They are $\frac{\partial f\left(x,y\right)}{dx}$ (for the partial derivative with respect to x) and $\frac{\partial f\left(x,y\right)}{dy}$ (for the partial derivative with respect to y). Using these partial derivatives, df_x and df_y can be expressed as

$\begin{array}{l}d{f}_x=\frac{\partial f}{\partial x}dx\\ d{f}_y=\frac{\partial f}{\partial y}dy\end{array}$

which then expresses the total differential of f (x, y) from Eq. 3.10 with respect to x and y as

$\begin{array}{l}df=d{f}_x+d{f}_y\hfill \\ =\frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy\hfill \end{array}$

For functions with more than two independent variables the total differential is calculated in the following form:

$df\left(x_1,x_2,\mathrm{...},x_n\right)=\frac{\partial f}{\partial x_1}d{x}_1+\frac{\partial f}{\partial x_2}d{x}_2+\mathrm{...}+\frac{\partial f}{\partial x_n}d{x}_n$

3.1.6.1 Product Rule

We often need to find the derivative of a function that contains a product. As an example, consider the function

$f\left(x\right)=g\left(x\right)h\left(x\right)$

In this case, the differential of f is found as

$\frac{df}{dx}=g\frac{dh}{dx}+h\frac{dg}{dx}$

Eq. 3.13 is referred to as the product rule. As a more practical example taken from fluid mechanics, the change of momentum along a coordinate axis is used in the derivation of the Navier-Stokes equation (see section 11). Here, the partial differential of the term $\rho \overrightarrow{v}$ with respect to x needs to be found. This term can be simplified as the following by using Eq. 3.13:

$\frac{\partial \left(\rho \overrightarrow{\upsilon }\right)}{\partial x}=\rho \frac{\partial \overrightarrow{\upsilon }}{\partial x}+\overrightarrow{\upsilon }\frac{\partial \rho }{\partial x}$

3.1.6.2 Chain Rule

Another important (and commonly employed) trick in differential calculus is the expansion of a (partial) derivative term in case the variables of a function are interdependent. As an example, if a function f (x, t) is dependent on x and t, it may be necessary to find the derivative $\frac{df\left(x,t\right)}{dt}$ of f (x, t) with respect to t. In some cases f depends primarily on x and only x depends on t. As an example, let

$f\left(x,t\right)=\frac{x^2-5}{t-2}$

be a function that depends on the variable x. Let x be time-variant with

$x\left(t\right)=e^{2t}$

Now inserting x (t) (which is a very easy term) into f (x, t) will allow finding $\frac{df}{dt}$, although this is still a somewhat tricky equation. Applying the chain rule of differentiation will make this solution much easier to find, yielding

$\frac{df}{dt}=\frac{df}{dx}\frac{dx}{dt}$

In this case the differential has been slightly rewritten and now requires the derivatives $\frac{df}{dx}$ and $\frac{dx}{dt}$ instead of only $\frac{df}{dt}$. These are very easy to find in the given case:

$\begin{array}{l}\frac{df}{dx}=\frac{2x}{t-2}\\ \frac{dx}{dt}=2e^{2t}\end{array}$

In which case we find

$\frac{df}{dt}=\frac{df}{dx}\frac{dx}{dt}=\frac{2x}{t-2}2e^{2t}$

This trick is referred to as a special form of the chain rule of differentiation which can be used in case of such interdependent variables. Note that these types of expansion of differentials can also be used on total differentials.

3.1.6.3 Integration by Parts

In contrast to the product rule for differentiation, integrating a product is a bit more complicated. Assume we have the following product of two functions: f (x) and g (x). We find the differential to be

$\frac{d\left(fg\right)}{dx}=f\frac{dg}{dx}+g\frac{df}{dx}$

Integration on both sides results in

$\begin{array}{ll}{\displaystyle \int \frac{d\left(fg\right)}{dx}}dx\hfill & ={\displaystyle \int f\frac{dg}{dx}}dx+{\displaystyle \int g\frac{df}{dx}}dx\hfill \\ {\displaystyle \int d\left(fg\right)}\hfill & ={\displaystyle \int f\frac{dg}{dx}}dx+{\displaystyle \int g\frac{df}{dx}}dx\hfill \\ fg\hfill & ={\displaystyle \int f\frac{dg}{dx}}dx+{\displaystyle \int g\frac{df}{dx}}dx\hfill \\ {\displaystyle \int f\frac{dg}{dx}}dx\hfill & =fg-{\displaystyle \int g\frac{df}{dx}}dx\hfill \end{array}$

Eq. 3.15 is referred to as integration by parts because the expression still contains integrals.

Example. Assume we have the following function for which we need to find the integral over a circular area:

${\displaystyle \int \frac{1}{r}}\frac{dc}{dr}dr$

where c (r) is a function of the radius r. Now, referring to Eq. 3.15, we define $f=\frac{1}{r}$ and $\frac{dg}{dx}=\frac{dc}{dr}$. This definition is arbitrary; we could have also chosen it the other way around. However, as we will see, this is the better approach. Using Eq. 3.15 we note that

$\begin{array}{l}f\left(r\right)=\frac{1}{r}\to \frac{df}{dr}=-\frac{1}{r^2}\\ \frac{dg}{dr}=\frac{dc}{dr}\to g\left(r\right)=c\left(r\right)\end{array}$

with which we can define the integral according to Eq. 3.15 as

${\displaystyle \int \frac{1}{r}}\frac{dc}{dr}dr=\frac{c}{r}+{\displaystyle \int \frac{c}{r^2}}dr$

which only leaves the integral $\frac{c}{r^2}dr$ to be evaluated.

3.1.7.1 Boundary Value Problems

A differential equation for which boundary conditions are given is commonly referred to as a boundary value problem (BVP).

How Many Boundary Conditions Do We Need? Often, we may know quite a number of things about a physical problem. From this knowledge we may deduce several boundary conditions. The question arises: How many boundary conditions do we need? In general, for each differential equation we need a number of boundary conditions corresponding to the degree of the differential equation. For a second-order differential equation we would require two boundary conditions; for a first-order differential equation we would require one.

3.1.7.2 Types of Boundary Conditions

There are usually two types of boundary conditions which we may come across. The first type indicates the value of the solution on the boundary of the domain.

Dirichlet Boundary Conditions. The Dirichlet¹boundary conditions state the value that the solution function f to the differential equation must have on the boundary of the domain C. The boundary is usually denoted as ∂C. In a two-dimensional domain that is described by x and y, a typical Dirichlet boundary condition would be

$f\left(x,y\right)=g\left(x,y,\mathrm{...}\right),\text{where:}\left(x,y\right)\in \partial C$

Here the function g may not only depend on x and y, but also on additional independent variables, e.g., the time t. Our example of the barrier lake used a Dirichlet boundary condition stating that the volume of the lake was 0 for t = 0. Here the function g is a constant, but it must not necessarily be the case.

Neumann Boundary Conditions. The second important type of boundary conditions are Neumann²boundary conditions. Neumann boundary conditions state that the derivative of the solution function f to the differential equation must have a given value on the boundary of the domain C. A typical Neumann boundary condition would be

$\frac{\partial f\left(x,y\right)}{\partial x}=g\left(x,y,\mathrm{...}\right)\text{where:}\left(x,y\right)\in \partial C$

or

$\frac{\partial f\left(x,y\right)}{\partial y}=g\left(x,y,\mathrm{...}\right)\text{where:}\left(x,y\right)\in \partial C$

Again the function g may not only depend on x and y, but also on additional variables such as time. We use an example of a Neumann boundary condition when deriving the velocity profile of a fluid layer on a plane driven by gravity (see section 15.3). While solving the differential equation we state that the shear stress on the open surface of the fluid must be 0 (see Eq. 15.12). This is the so-called free-surface condition (see section 9.8.3). As the shear stress is calculated from the first derivative of the velocity, this is a Neumann boundary condition.

3.2.1.1 Sine Function

The sine function sin x is periodic over the period length T = 2π (see Fig. 3.4a). It is point-symmetric to the origin and is therefore referred to as an odd function. It obtains specific values for

$\begin{array}{ll}\text{all}x=n\pi \hfill & \sin x=0\hfill \\ \text{all}x=\left(4n+1\right)\frac{\pi }{2}\hfill & \sin x=1\hfill \\ \text{all}x=\left(4n-1\right)\frac{\pi }{2}\hfill & \sin x=-1\hfill \end{array}$

As the sine function is point-symmetric to the origin we note that

$\sin \left(-x\right)=-\sin x$

$\sin \left(n\pi +x\right)=\sin \left(n\pi -x\right)\text{for all odd values of }n$

$\sin \left(n\pi +x\right)=-\sin \left(n\pi -x\right)\text{for all even values of }n$

3.2.1.2 Cosine Function

The cosine function cos x is also periodic over the period length T = 2π (see Fig. 3.4a). It is mirror-symmetric to x = 0 and is therefore referred to as an even function. It obtains specific values for

$\begin{array}{ll}\text{all }x=2n\pi \hfill & \cos x=1\hfill \\ \text{all}x=\left(2n+1\right)\pi \hfill & \cos x=-1\hfill \\ \text{all}x=\left(2n+1\right)\frac{\pi }{2}\hfill & \cos x=0\hfill \end{array}$

As the cosine function is mirror-symmetric to the origin we note that

$\cos \left(-x\right)=\cos x$

$\cos \left(n\pi +x\right)=\cos \left(n\pi -x\right)\text{for all even values of }n$

$\cos \left(n\pi +x\right)=-\cos \left(n\pi -x\right)\text{for all odd values of }n$

3.2.1.3 Tangent Function

The tangent function tan x is also periodic over the period length T = 2π (see Fig. 3.4a). It is defined as

$\tan x=\frac{\sin x}{\cos x}$

It is point-symmetric to the origin and therefore also an odd function. It obtains specific values for

$\begin{array}{ll}·\text{all}x=n\pi \hfill & \tan x=0\hfill \\ ·\text{all}x=(2n+1)\frac{\pi }{2}\hfill & \text{tan }x=\pm \infty \hfill \end{array}$

As you can see, the tangent function is discontinuous for all values of $x=\left(2n+1\right)\frac{\pi }{2}$

3.2.1.4 Hyperbolic Sine Function

The hyperbolic sine function sinh x is not periodic (see Fig. 3.4b). It is point-symmetric to the origin and therefore also an odd function. It is defined as

$\sinh x=\frac{1}{2}\left({\text{e}}^x-{\text{e}}^{-x}\right)$

The most important points to note are the fact that sinh (x = 0) = 0, and it does not obtain the value of 0 for any other values of x.

3.2.1.5 Hyperbolic Cosine Function

The hyperbolic cosine function cosh x is also nonperiodic (see Fig. 3.4b). It is mirror-symmetric to x = 0 and therefore also an even function. It is defined as $\cosh x=\frac{1}{2}\left({\text{e}}^x+{\text{e}}^{-x}\right)$

The most important point is the fact that cosh (x = 0) = 1, and it never obtains the value of 0 for any value of x.

3.2.1.6 Hyperbolic Tangent Function

The hyperbolic tangent function tanh x is also nonperiodic (see Fig. 3.4b) and defined, in analogy to the regular tangent function, as

$\tanh x=\frac{\sinh x}{\cosh x}$

It is point-symmetric to the origin and therefore also an add function. It obtains specific values for

• tanh (x = 0) = 1

• tanh x → 1 for x > 2

• tanh x → −1 for x < −2

Occasionally, you may come across the hyperbolic cotangent function, which is defined as

$\coth x=\frac{\cosh x}{\sinh x}$

3.2.1.7 Derivatives

In the following, we will note a couple of important derivatives. The derivatives of sine and cosine are cyclic, yielding

$\begin{array}{ll}\frac{d}{dx}\sin x\hfill & =\cos x\hfill \\ \frac{d}{dx}\cos x\hfill & =-\sin x\hfill \\ \frac{d}{dx}\left(-\sin x\right)\hfill & =-\cos x\hfill \\ \frac{d}{dx}\left(-\cos x\right)\hfill & =\sin x\hfill \end{array}$

The derivative of the tangent function is

$\begin{array}{l}\frac{d}{dx}\tan x=\frac{1}{{\cos }^{2}x}\hfill \\ \frac{d}{dx}\cot x=-\frac{1}{{\sin }^{2}x}\hfill \end{array}$

The derivatives of the hyperbolic functions are

$\begin{array}{l}\frac{d}{dx}\sinh x=\cosh x\hfill \\ \frac{d}{dx}\cosh x=\sinh x\hfill \\ \frac{d}{dx}\tanh x=\frac{1}{{\cosh }^{2}x}\hfill \\ \frac{d}{dx}\coth x=-\frac{1}{{\sinh }^{2}x}\hfill \end{array}$

3.2.2.1 Pythagorean Theorem

More often than expected, the Pythagorean theorem is used, which states

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

3.2.2.2 Inverse Functions

As a short reminder, each trigonometric function has an inverse function that is defined as

$\begin{array}{ll}\text{sine}:\hfill & \sin x=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{1}{\csc x}\hfill \\ \text{cosecant}:\hfill & \text{scs}x=\frac{\text{hypotenuse}}{\text{opposite}}=\frac{1}{\sin x}\hfill \\ \text{cosine:}\hfill & \text{cos}x=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{1}{\sec x}\hfill \\ \text{secant}:\hfill & \text{sec}x=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{1}{\cos x}\hfill \\ \text{tangent}:\hfill & \tan x=\frac{\text{opposite}}{\text{adjacent}}=\frac{1}{\tan x}\hfill \\ \text{cotangent:}\hfill & \cot x=\frac{\text{adjacent}}{\text{opposite}}=\frac{1}{\tan x}\hfill \end{array}$

3.2.2.3 Conversion From Sine

The following equations allow expressing the other trigonometric functions by means of the sine function:

$\sin x=\pm \sqrt{1-{\cos }^{2}x}$

$=\pm \frac{\tan x}{\sqrt{1+{\tan }^{2}x}}$

$=\pm \frac{1}{\sqrt{1+{\cot }^{2}x}}$

3.2.2.4 Conversion From Cosine

The following equations allow expressing the other trigonometric functions by means of the cosine function:

$\cos x=\pm \sqrt{1-{\sin }^{2}x}$

$=\pm \frac{1}{\sqrt{1+{\tan }^{2}x}}$

$=\pm \frac{\cot x}{\sqrt{1+{\cot }^{2}x}}$

3.2.2.5 Conversion From Tangent

The following equations allow expressing the other trigonometric functions by means of the tangent function:

$\tan x=\pm \frac{\sin x}{\sqrt{1-{\sin }^{2}x}}$

$=\pm \frac{\sqrt{1-{\cos }^{2}x}}{\cos x}$

$=\pm \frac{1}{\cot x}$

3.2.2.6 Conversion From Cotangent

The following equations allow expressing the other trigonometric functions by means of the cotangent function:

$\cot x=\pm \frac{\sqrt{1-{\sin }^{2}x}}{\sin x}$

$=\pm \frac{\cos x}{\sqrt{1-{\cos }^{2}x}}$

$=\pm \frac{1}{\tan x}$

3.2.2.7 Conversions For Half-Angles

The following equations can be used in order to convert half-angles into equations of full angles:

$\sin \frac{x}{2}=\pm \frac{\sqrt{1-\cos x}}{2}$

$\cos \frac{x}{2}=\pm \frac{\sqrt{1+\cos x}}{2}$

$\tan \frac{x}{2}=\pm \frac{\sqrt{1-\cos x}}{1+\cos x}=\frac{\sin x}{1+\cos x}=\frac{1-\cos x}{\sin x}$

3.2.4 Bessel Functions

3.2.4.1 Standard Bessel Functions

The Bessel² functions were first described by German mathematician Friedrich Wilhelm Bessel while studying many-body problems, particularly planetary motion [7]. Bessel was not the first to describe these equations, but he was the first to extensively use them to solve partial differential equations.

Definition.Bessel functions are solutions to a specific type of differential equation referred to as Bessel’s differential equation. This differential equation is a special case of the a Sturm-Liouville boundary value problem, which often occurs in polar and cylindrical coordinates. The Bessel differential equation of order ν for the function f (x) is given by

$x^2\frac{d^{2}f\left(x\right)}{d{x}^2}+x\frac{df\left(x\right)}{dx}+\left({\alpha }^2{x}^2-v^2\right)f\left(x\right)=0$

In this equation, ν is a nonnegative integer. In general, ν could also be a real number but the solutions to these differential equations are somewhat different. Eq. 3.44 is a second-order ordinary differential equation and is solved by a set of two functions, the Bessel function of first kind J_ν (x) and the Bessel function of second kind Y_ν (x) which is also referred to as Weber function. The Bessel function of first kind J_ν (x) can be approximated by a series of gamma functions. The Bessel function of second kind Y_ν (x) can be expressed using the Bessel function of first order.

The first five Bessel functions J_ν (x) and Y_ν (x), respectively, are displayed in Fig. 3.6. As can be seen, the functions look roughly like decaying sine or cosine functions.

The general solution to Eq. 3.44 is

$f\left(x\right)=c_1{J}_v\left(\alpha x\right)+c_2{Y}_v\left(\alpha x\right)$

In physical applications, c₂ = 0 in most cases. This is due to the fact that Y_ν (α x) strives to negative infinity for x → 0, which yields physically unreasonable results (unless there is a very strong source or sink at x = 0). Therefore, the general solution to Eq. 3.44 in most practical physical problems is

$f\left(x\right)=c_1{J}_v\left(\alpha x\right)$

Important Properties. Some of the most important properties of the Bessel function are for J_ν

$\frac{d}{dx}\left(J_v\left(\alpha x\right)\right)=\alpha J_{v-1}\left(\alpha x\right)-\frac{v}{x}\alpha J_v\left(\alpha x\right)$

$\frac{d}{dx}\left(x^v{J}_v\left(\alpha x\right)\right)=\alpha x^v{J}_{v-1}\left(\alpha x\right)$

$\frac{d}{dx}\left(x^{-v}{J}_v\left(\alpha x\right)\right)=-x^{-v}\alpha J_{v+1}\left(\alpha x\right)$

$J_{v+1}\left(\alpha x\right)=\frac{2v}{\alpha x}{J}_v\left(\alpha x\right)-J_{v-1}\left(\alpha x\right)$

$\frac{d}{dx}\left(x^v{J}_v\left(\alpha x\right)\right)=x^{v-1}v{J}_v\left(\alpha x\right)+x^v\left(\frac{v}{x}{J}_v\left(\alpha x\right)-\alpha J_{v+1}\left(\alpha x\right)\right)$

$\frac{d}{dx}\left(J_v\left(\alpha x\right)\right)=\frac{v}{x}{J}_v\left(\alpha x\right)-\alpha J_{v+1}\left(\alpha x\right)$

$\frac{d}{dx}\left(x^{v+3}{J}_v\left(\alpha x\right)\right)-\frac{2}{\alpha }{x}^{v+1}{J}_{v+2}\left(\alpha x\right)=\alpha x^{v+3}{J}_v\left(\alpha x\right)$

$J_v\left(-\lambda x\right)={\left(-1\right)}^v{J}_v\left(\lambda x\right)$

for Y_ν

$\frac{d}{dx}\left(Y_v\left(\alpha x\right)\right)=\alpha Y_{v-1}\left(\alpha x\right)-\frac{v}{x}\alpha Y_v\left(\alpha x\right)$

$\frac{d}{dx}\left(x^v{Y}_v\left(\alpha x\right)\right)=x^v\alpha Y_{v-1}\left(\alpha x\right)$

which for ν = 0 gives the important relations

$\frac{d}{dx}\left(x{J}_1\left(\alpha x\right)\right)=\alpha x{J}_0\left(\alpha x\right)$

$\frac{d}{dx}\left(x{Y}_1\left(\alpha x\right)\right)=\alpha x{Y}_0\left(\alpha x\right)$

Here you can also verify one important property given by

$\frac{d{J}_0\left(x\right)}{dx}=-J_1\left(x\right)$

Recurrence Relation. Among the most important properties of Bessel functions are the so-called recurrence relations, which state that Bessel functions of higher order can be expressed by Bessel functions of lower order and v.v. The recurrence relation reads

$J_{v-1}\left(\alpha x\right)=\frac{2v}{\alpha x}{J}_v\left(\alpha x\right)-J_{v+1}\left(\alpha x\right)$

$Y_{v-1}\left(\alpha x\right)=\frac{2v}{\alpha x}{Y}_v\left(\alpha x\right)-Y_{v+1}\left(\alpha x\right)$

General Notation. In many applications the standard Bessel differential equation occurs in slightly different form. Compared to Eq. 3.44 a more general notation can be written as

$x^2\frac{d^{2}f\left(x\right)}{d{x}^2}+\left(2p+1\right)x\frac{df\left(x\right)}{dx}+\left({\alpha }^2{x}^{2r}+v^2\right)f\left(x\right)=0$

The general solution to Eq. 3.44 is

$\begin{array}{l}f\left(x\right)=x^{-p}\left(c_1{J}_{\frac{q}{r}}\left(\frac{\alpha }{r}{x}^r\right)+c_2{Y}_{\frac{q}{r}}\left(\frac{\alpha }{r}{x}^r\right)\right)\hfill \\ \text{with}q=\sqrt{p^2-v^2}\hfill \end{array}$

where, again, the second term is often dropped in order to obtain physically meaningful results.

Series Notation of the Bessel Functions of First Kind. The Bessel function of first kind is often written out explicitly as a series that is given by

$J_v\left(x\right)={\displaystyle \sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{n!\left(v+1\right)!}{\left(\frac{x}{2}\right)}^{v+2n}}$

Now this notation does not look very convenient, but it is simple to write it out for explicit values of ν. The first two Bessel functions for ν = 0 and ν = 1 can be written out as

$J_0\left(x\right)=1-{\left(\frac{x}{2}\right)}^2+\frac{1}{{\left(2!\right)}^2}{\left(\frac{x}{2}\right)}^4-\frac{1}{{\left(3!\right)}^2}{\left(\frac{x}{2}\right)}^6+\cdots$

$J_1\left(x\right)=\frac{x}{2}-\frac{1}{2!}{\left(\frac{x}{2}\right)}^3+\frac{1}{2!3!}{\left(\frac{x}{2}\right)}^5-\frac{1}{3!4!}{\left(\frac{x}{2}\right)}^7+\cdots$

which looks distinctively more common.

Gamma Function. In these derivations we have always assumed ν to be an integer, which it does not necessarily have to be. If we allow all real values for ν the series notation of the Bessel function of first kind can be upheld, but it is rewritten using the so-called gamma function Γ which is defined as

$\Gamma \left(x\right)={\int }_0^{\infty }{t}^{x-1}{e}^{-t}dt\left(x>0\right)$

Fig. 3.7 shows a plot of the gamma function Γ (x). The gamma function is an important function in integral and differential analysis and can be used to express the Bessel function of first kind for all ν including noninteger values as

$J_v\left(x\right)={\displaystyle \sum _{n=0}^{\infty }\frac{{\left(-1\right)}^n}{n!\Gamma \left(v+n+1\right)!}{\left(\frac{x}{2}\right)}^{v+2n}}$

3.2.4.2 Modified Bessel Functions

Definition. If Eq. 3.44 is used with complex numbers for the dependent variable x the resulting differential equation is referred to the modified Bessel’s differential equation. Therefore, modified Bessel functions are solutions to the modified Bessel’s differential equation of order ν which, for the function f (x), is given by

$x^2\frac{d^{2}f\left(x\right)}{d{x}^2}+x\frac{df\left(x\right)}{dx}-\left({\left(\alpha x\right)}^2+v^2\right)f\left(x\right)=0$

where α is a constant. The general solution to Eq. 3.68 is given by a set of two functions, the modified Bessel function of first kind I_ν (α x) and the modified Bessel function of second kind K_ν (α x). The solution is given by

$f\left(x\right)=c_1{I}_v\left(\alpha x\right)+c_2{K}_v\left(\alpha x\right)$

For the same reason as for the standard Bessel functions c₂ = 0 in order to yield physically meaningful results, which simplifies the solution to

$f\left(x\right)=c_1{I}_v\left(\alpha x\right)$

The first five Bessel functions I_ν (x) and K_ν (x), respectively, are displayed in Fig. 3.8.

Important Properties.I_ν (x) can be expressed by the standard Bessel functions of first kind as

$I_v\left(x\right)={\text{i}}^{-v}{J}_v\left(ix\right)$

Likewise, K_ν (x) can be expressed by the standard Bessel functions of first and second kind as

$K_v\left(x\right)=\frac{\pi }{2}{\text{i}}^{v+1}\left(J_v\left(\text{i}x\right)+\text{i}{Y}_v\left(\text{i}x\right)\right)$

Similar to the standard Bessel functions, modified Bessel functions also feature recurrence relationships. Therefore, derivatives of higher Bessel functions can be expressed by Bessel functions of lower orders. The general relationship is the following:

for I_ν

$\frac{d}{dx}\left(I_v\left(\alpha x\right)\right)=\alpha I_{v-1}\left(\alpha x\right)-\frac{v}{x}\alpha I_v\left(\alpha x\right)$

$\frac{d}{dx}\left(x^v{I}_v\left(\alpha x\right)\right)=x^v{I}_{v-1}\left(\alpha x\right)$

for K_ν

$\frac{d}{dx}\left(K_v\left(\alpha x\right)\right)=-\alpha K_{v-1}\left(\alpha x\right)-\frac{v}{x}\alpha K_v\left(\alpha x\right)$

$\frac{d}{dx}\left(x^v{K}_v\left(\alpha x\right)\right)=x^v{K}_{v-1}\left(\alpha x\right)$

which for ν = 0 gives the important relations

$\frac{d}{dx}{I}_0\left(\alpha x\right)=\alpha I_1\left(\alpha x\right)$

$\frac{d}{dx}{K}_0\left(\alpha x\right)=-\alpha K_1\left(\alpha x\right)$

Recurrence Relation. Just like the regular Bessel functions the modified Bessel functions also show recurrence relation:

$J_{v-1}\left(\alpha x\right)=-\frac{2v}{\alpha x}{J}_v\left(\alpha x\right)+J_{v+1}\left(\alpha x\right)$

$Y_{v-1}\left(\alpha x\right)=-\frac{2v}{\alpha x}{Y}_v\left(\alpha x\right)+Y_{v+1}\left(\alpha x\right)$

3.2.4.3 Roots of the Bessel Functions

As you can see from Fig. 3.6 the Bessel function of first and second kind have a number of roots, i.e., values for x_n at which J_ν (x_n) = 0 and Y_ν (x_n) = 0. As it turns out, these values are very important and frequently used when dealing with Bessel functions and in particular with Bessel differential equations. Since the roots are simply numbers, they can be calculated conveniently. Tab. 3.1 lists the first ten roots of the first five Bessel functions of first and second kind, respectively.